part i chapter 1 functions and their graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m =...

86
CHAPTER 1 Functions and Their Graphs Section 1.1 Lines in the Plane . . . . . . . . . . . . . . . . . . . . . . . 2 Section 1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Section 1.3 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . 24 Section 1.4 Shifting, Reflecting, and Stretching Graphs . . . . . . . . . 35 Section 1.5 Combinations of Functions . . . . . . . . . . . . . . . . . 43 Section 1.6 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . 54 Section 1.7 Linear Models and Scatter Plots . . . . . . . . . . . . . . . 68 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 PART I © Houghton Mifflin Company. All rights reserved.

Upload: others

Post on 31-Dec-2020

2 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

C H A P T E R 1Functions and Their Graphs

Section 1.1 Lines in the Plane . . . . . . . . . . . . . . . . . . . . . . . 2

Section 1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Section 1.3 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . 24

Section 1.4 Shifting, Reflecting, and Stretching Graphs . . . . . . . . . 35

Section 1.5 Combinations of Functions . . . . . . . . . . . . . . . . . 43

Section 1.6 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . 54

Section 1.7 Linear Models and Scatter Plots . . . . . . . . . . . . . . . 68

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

P A R T I©

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.

Page 2: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

3.

2

2

4

4

6

6

8

8

10

10

(2, 3)

x

y

m = −3m = 1

m = 2

m = 0

4.

–6 –2

–4

–2

2

4

(−4, 1)

x

y

m =

m = −3 m = 3

12

m is undefined.

C H A P T E R 1Functions and Their Graphs

Section 1.1 Lines in the Plane

2

You should know the following important facts about lines.

■ The graph of is a straight line. It is called a linear equation.

■ The slope of the line through and is

■ (a) If the line rises from left to right. (b) If the line is horizontal.

(c) If , the line falls from left to right. (d) If m is undefined, the line is vertical.

■ Equations of Lines

(a) Slope-Intercept: (b) Point-Slope:

(c) Two-Point: (d) General:

(e) Vertical: (f ) Horizontal:

■ Given two distinct nonvertical lines

(a) is parallel to if and only if

(b) is perpendicular to if and only if m1 � �1�m2.L2L1

m1 � m2 and b1 � b2.L2L1

L1: y � m1x � b1 and L2: y � m2x � b2

y � bx � a

Ax � By � c � 0y � y1 �y2 � y1

x2 � x1

�x � x1�

y � y1 � m�x � x1�y � mx � b

m < 0,

m � 0,m > 0

m �y2 � y1

x2 � x1

.

�x2, y2��x1, y1�y � mx � b

1. (a) Since the slope is positive, the line rises.Matches

(b) m is undefined. The line is vertical. Matches

(c) The line falls. Matches L1.m � �2.

L3.

L2.m �

23. 2. (a) The line is horizontal. Matches

(b) Because the slope is negative, the linefalls. Matches

(c) Because the slope is positive, the linerises. Matches L3.m � 1.

L1.m � � 3

4.

L2.m � 0.

Vocabulary Check

1. (a) iii (b) i (c) v (d) ii (e) iv 2. slope

3. parallel 4. perpendicular 5. linear extrapolation

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 3: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.1 Lines in the Plane 3

5. Slope �rise

run�

3

26. The line appears to go through and

Slope �8 � 00 � 2

� �4

�2, 0�.�0, 8�

7.

−12

−12 12

4

(−4, 0)

(0, −10)

Slope �0 � ��10�

�4 � 0�

10�4

� �52

8.

−6

−6

12

6

(2, 4)

(4, −4)

Slope ��4 � 4

4 � 2� �4

9.

Slope is undefined.

−2

−10 2

6

(−6, 4)

(−6, −1)

10.

−10

−3

5

7

(1, 6)

(−3, −2)

Slope �6 � ��2�1 � ��3� �

84

� 2

11. Since y does not change. Three points are�0, 1�, �3, 1�, and ��1, 1�.

m � 0, 12. Since y does not change. Three additionalpoints: �4, �2�.�1, �2�,�0, �2�,

m � 0,

14. Because m is undefined, x does not change. Threeother points are: ��4, 5�.��4, 3�,��4, 0�,

13. Since m is undefined, x does not change and the lineis vertical. Three points are and �1, 3�.�1, 2�,�1, 1�,

15. Since y decreases 2 for every unitincrease in x. Three points are and �3, �15�.

�2, �13�,�1, �11�,m � �2, 16. Since y increases 2 for every unit increase

in x. Three points are: and��2, 10�.

��3, 8�,��4, 6�,m � 2,

17. Since increases 1 for every increase of 2in Three points are and �13, 1�.�9, �1�, �11, 0�,x.

m �12, y 18. Since y decreases 1 for every increase of

2 units in x. Three points are �5, �9�.

�3, �8�,�1, �7�,m � �

12,

19.

(a) Slope:

y-intercept:

(b)

–4 –3 –2 –1 1 2

3

4

5

(0, 3)

x

y

�0, 3�

m � 5

y � 5x � 3

5x � y � 3 � 0 20.

(a) Slope:

y-intercept:

(b)

–1 1 2 3 4

1

2

4

5

x

(0, 3)

y

�0, 3�

m � �23

y � �23 x � 3

3y � �2x � 9

2x � 3y � 9 � 0

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 4: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

4 Chapter 1 Functions and Their Graphs4 Chapter 1 Functions and Their Graphs

21.

(a) Slope: undefined

No y-intercept

(b)

–1 1 2 3

–2

–1

1

2

x

y

x �25

5x � 2 � 0 22.

(a) Slope: undefined

y-intercept: none

(b)3

2

1

−1

−2

−3

321−1−3x

y

x � �73

3x � 7 � 0 23.

(a) Slope:

-intercept:

(b)

x

53( )

−2 −1 1 2

−1

−2

−3

1

0, −

y

�0, �53�y

m � 0

y � �53

3y � 5 � 0

24.

(a) Slope:

y-intercept:

(b)

0, − ))

3

2

1

−1

−2

−3

321−1−2−3x

y

118

�0, �118 �

m � 0

y � �118

8y � �11

�11 � 8y � 0 25.

–2 –1 1 2 3 4

–2

–1

1

2

x

(0, −2)

y

� 2y � 3x � 2 ⇒ 3x � y � 2 � 0

y � 2 � 3�x � 0�

26. (a)

–6 –4 –2 2 4 6

–6

–4

4

6

x

(−3, 6)

y

2x � y � 0

y � �2x

y � 6 � �2�x � 3�

m � �2, ��3, 6� 27.

–2 –1 1 2 3 4

–3

–4

–5

–1

1

x

(2, −3)

y

x � 2y � 4 � 0

2y � 4 � �x

y � 3 � �12 x � 1

y � ��3� � �12�x � 2� 28.

–2 2

–2

x

(−2, −5)

y

0 � 3x � 4y � 14

4y � 20 � 3x � 6

y � 5 �34�x � 2�

m �34, ��2, �5�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 5: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.1 Lines in the Plane 5

29.

–4 –2 2 4

–6

–4

–2

2

4

6

x(6, −1)

y

x � 6 � 0

x � 6 30. m undefined. Line is vertical.

x

y

−12 −8 −4 4

−4

−8

8

4

x � 10 � 0

31.

horizontal line

3

4

2

1

−1

−2

321−1−2−3

− ,( (

x

y

12

32

y �32 � 0

y �32 � 0�x �

12�

32. Line is horizontal.

x

y

−8 −6 −4 −2 4 8 102 6

−4−6

−10

−14−12

−16

2

4

(2.3, −8.5)

y � 8.5 � 0

m � 0. 33.

−1

−2 4

3

1 � y � �3

5x � 2

� 1y � �3

5�x � 5� � 1

y � 1 �5 � 1

�5 � 5�x � 5� 34.

−6

−4

6

4

y �78

x �12

y � 3 �7

8�x � 4�

y � 3 ��4 � 3

�4 � 4�x � 4�

�4, 3�, ��4, �4�

35. Since both points havethe slope is

undefined.

−4

−10 2

4

x � �8

x � �8,36.

−6

−2

6

6

y � 4

y � 4 � 0

y � 4 � 0�x � 1�

y � 4 �4 � 4

6 � ��1��x � 1�

��1, 4�, �6, 4� 37.

−1

−2 4

3

y � �1

2x �

3

2

y � �1

2�x � 2� �

1

2

y �1

2�

54 � 1

212 � 2

�x � 2�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 6: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

6 Chapter 1 Functions and Their Graphs6 Chapter 1 Functions and Their Graphs

38.

−6

−3

6

5

y � �13

x �43

y � 1 � �1

3x �

1

3

y � 1 � �1

3�x � 1�

y � 1 �� 2

3 � 1

6 � 1�x � 1�

�1, 1�, �6, �2

3� 39.

−6

−9 9

6

y � �65

x �1825

y �35

� �65�x �

110�

y �35

��

95 �

35

910 �

110�x �

110�

40.

−3

−1

3

3

y � �3

25x �

159100

y �32

� �325

x �9

100

y �32

� �325 �x �

34�

y �32

�74 �

32

�43 �

34�x �

34�

�34

, 32�, ��

43

, 74� 41.

−2

−3 3

2

� 0.6y � 0.4x � 0.2

� 0.6y � 0.4�x � 1� � 0.6

y � 0.6 ��0.6 � 0.6

�2 � 1�x � 1� 42.

−6

−5

6

3

y � �310

x �95

y � 0.6 � �3

10�x � 8�

y � 0.6 ��2.4 � 0.6

2 � ��8��x � 8�

��8, 0.6�, �2, �2.4�

43. The slope is

y � 2x � 5

y � 3 � 2x � 2

y � ��3� � 2�x � 1�

�3 � ��7�1 � ��1� �

42

� 2. 44. The slope is

y � �12

x � 1

y � 1 � �12

x � 2

y � ��1� � �12

�x � 4�

�1 �32

4 � ��1� ��

52

5�

�12

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.

Page 7: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.1 Lines in the Plane 7

45. Using the points and you have

S � 2200�2008� � 4,380,300 � $37,300.

When t � 2008,

S � 2200t � 4,380,300.

S � 28,500 � 2200�t � 2004�

m �32,900 � 28,500

2006 � 2004�

44002

� 2200

�2006, 32,900�,�2004, 28,500� 46. Using the points and you have

S � 1250�2008� � 2,480,000 � $30,000.

When t � 2008,

S � 1250t � 2,480,000.

S � 25,000 � 1250�t � 2004�

m �27,500 � 25,000

2006 � 2004�

25002

� 1250

�2006, 27,500�,�2004, 25,000�

47.

Slope:

y-intercept:

The graph passes through and rises 1 unitfor each horizontal increase of 2.

�0, �2�

�0, �2�

12

y �12

x � 2

�2y � �x � 4

x � 2y � 4 48.

Slope:

y-intercept:

The line slopes downward and passes through thepoint �0, 14�.

�0, 14�

�34

y ��34

x �14

4y � �3x � 1

3x � 4y � 1

49.

slope is undefined

no y-intercept

The line is vertical and passes through ��6, 0�.

x � �6 50.

Slope: 0

y-intercept:

The line is horizontal and passes through �0, 12�.

�0, 12�

y � 12

51.

The second setting shows the x- and y-intercepts more clearly.

−4

−2 10

1

−1

−5 10

10

y � 0.5x � 3 52.

The first setting shows the x- and y-intercepts more clearly.

−5

−80

10

80

−5

−10

5

10

53.

and are perpendicular.L2L1

mL2�

1 � 3

4 � 0� �

1

2� �

1

mL1

−4

−12 12

12

(0, 3)

(0, −1)

(4, 1)

(5, 9)

mL1�

9 � 1

5 � 0� 2

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 8: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

8 Chapter 1 Functions and Their Graphs8 Chapter 1 Functions and Their Graphs

54.

The lines are neither parallel nor perpendicular.

m2 ��5 � 3

5 � 1�

�8

4� �2

L2: �1, 3�, �5, �5�

m1 �5 � ��1�1 � ��2�

�6

3� 2 −12

−8

12

8

(1, 5)(1, 3)

(−2, −1)

(5, −5)

L1: ��2, �1�, �1, 5� 55.

and are parallel.L2L1

mL2�

73 � 1

5 � 0�

2

3� mL1

−8

−12 12

8

(0, −1)

(−6, 0)

(3, 6)

5, 73( (

mL1�

0 � 6

�6 � 3�

2

3

56.

The lines are perpendicular.

−12

−6

12

10

(−4, 2)

(4, 8)

(3, −5)−1, )( 1

3

m2 ��1�3� � ��5�

�1 � 3�

16�3

�4� �

4

3

L2: �3, �5�, ��1, 1

3�

m1 �2 � 8

�4 � 4�

�6

�8�

3

4

L1: �4, 8�, ��4, 2� 57.

Slope:

(a)

(b)

1 � y � �1

2 x � 2

y � 1 � �1

2 �x � 2�

1 � y � 2x � 3

y � 1 � 2�x � 2�

m � 2

y � 2x �3

2

4x � 2y � 3

58.

Slope:

(a)

(b)

y � x � 5

y � 2 � 1�x � 3�

m � 1, ��3, 2�

y � �x � 1

y � 2 � �1�x � 3�

m � �1, ��3, 2�

m � �1

y � �x � 7

x � y � 7 59.

Slope:

(a)

(b)

y �43x �

12772

y �78 �

43�x �

23�

y � �34x �

38

y �78 � �

34�x �

23�

m � �34

y � �34x �

74

3x � 4y � 7 60.

Slope:

(a)

(b)

y �13 x �

110

y � 1.4 �13 �x � 3.9�

m �13 , ��3.9, �1.4�

y � �3x � 13.1

y � 1.4 � �3�x � 3.9�

m � �3, ��3.9, �1.4�

m � �3

y � �3x �92

2y � �6x � 9

6x � 2y � 9

61. vertical line

slope not defined

(a) passes through

(b) passes through and ishorizontal

�3, �2�y � �2

�3, �2�x � 3 � 0

x � 4 � 0 62.

Slope:

(a)

(b) m undefined (vertical line)

x � �4

y � 1

m � 0, ��4, 1�

m � 0

y � �2

y � 2 � 0

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 9: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.1 Lines in the Plane 9

63. The slope is 2 and lies on the line. Hence,

y � 2x � 1.

y � 1 � 2�x � 1�

y � ��1� � 2�x � ��1��

��1, �1� 64. The slope is and lies on the line. Hence,

y � �2x � 1.

y � 1 � �2�x � 1�

y � 1 � �2�x � ��1��

��1, 1��2

65. The slope of the given line is 2. Then has slope Hence,

y � �12x � 1.

y � 2 � �12�x � 2�

y � 2 � �12�x � ��2��

�12.l 66. The slope of the given line is 3. Then has slope

Hence,

y � �13x � 4.

y � 5 � �13�x � 3�

y � 5 � �13�x � ��3��

�13.l

67. (a) (b) (c)

(b) and (c) are perpendicular.

−10

−15 15

10y = 2x

y = − 2xy = x12

y �12xy � �2x y � 2x 68.

is parallel to is perpendicular to and L3.

L1L2L3.L1

−9

−6

9

6y = x + 22

3

y = x23

y = − x32

L3: y �23x � 2L2: y � �3

2x;L1: y �23x;

69. (a) (b)

(c)

(a) and (b) are parallel.

(c) is perpendicular to (a) and (b).

−10

−15 15

10y = 2x − 4

y = − x12

y = − x + 312

y � 2x � 4

y � �12x � 3y � �

12x 70.

is parallel to is perpendicular to and L2.L1L3L2.L1

−15

−10

15

10

y = −x + 3

y = x + 1

y = x − 8

L3: y � �x � 3L2: y � x � 1;L1: y � x � 8;

71. (a) Years Slope

1995–1996

1996–1997

1997–1998

1998–1999

1999–2000

2000–2001

2001–2002

2002–2003

2003–2004

Greatest increase: 1998–1999

Greatest decrease: 1999–2000 ��0.78��0.86�

0.31 � 0.00 � 0.31

0.00 � 0.20 � �0.20

0.20 � 0.92 � �0.72

0.92 � 0.82 � 0.10

0.82 � 1.60 � �0.78

1.60 � 0.74 � 0.86

0.74 � 0.57 � 0.17

0.57 � 0.69 � �0.12

0.69 � 0.91 � �0.22

(b)

(c) Between 1995 and 2004, the earnings per sharedecreased at the rate of 0.07 per year.

(d) For 2010, andwhich is

reasonable.y � �0.07�20� � 1.24 � �0.16,

x � 20

y � �0.07x � 1.24

y � �115

�x � 5� �91100

� �1

15x �

373300

y � 0.91 �0.31 � 0.91

14 � 5 �x � 5�

�14, 0.31�:�5, 0.91�,

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 10: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

10 Chapter 1 Functions and Their Graphs10 Chapter 1 Functions and Their Graphs

72. (a) Years Slope

1995–1996

1996–1997

1997–1998

1998–1999

1999–2000

2000–2001

2001–2002

2002–2003

2003–2004

Greatest increase: 2003–2004

Smallest increase: 1996–1997 �0.1��3.3�

18.4 � 15.1 � 3.3

15.1 � 13.9 � 1.2

13.9 � 14.1 � �0.2

14.1 � 14.4 � �0.3

14.4 � 12.9 � 1.5

12.9 � 12.6 � 0.3

12.6 � 13.2 � �0.6

13.2 � 13.1 � 0.1

13.1 � 13.2 � �0.1

(b)

(c) Between 1995 and 2004, the sales (in billions ofdollars) increased at the rate of 0.58 per year.

(d) For 2010, and(billion), which

seems reasonable.y � 0.58�20� � 10.31 � 21.91

x � 20

y � 0.58x � 10.31

y �2645

x �46445

y � 13.2 �18.4 � 13.2

14 � 5�x � 5�

�5, 13.2�, �14, 18.4�

73.

The maximum height in the attic is 12 feet.

x � 12

4x � 48

34

�x

16

riserun

�34

�x

12�32� 74.

x � 16,66623 ft � 3.16 miles

�12x � ��2000��100�

�12100

��2000

x

Slope �riserun

75.

V � 125t � 1790

V � 2540 � 125�t � 6�

�6, 2540�, m � 125 76.

V � 4.50t � 129

V � 156 � 4.50�t � 6�

�6, 156�, m � 4.50

77.

V � �2000t � 32,400

V � 20,400 � �2000�t � 6�

�6, 20,400�, m � �2000 78.

V � �5600t � 278,600

V � 245,000 � �5600�t � 6�

�6, 245,000�, m � �5600

79. The slope is This represents the decreasein the amount of the loan each week. Matches graph (b).

m � �10. 80. The y-intercept is 12.5 and the slope is 1.5, whichrepresents the increase in hourly wage per unitproduced. Matches graph (c).

81. The slope is This represents the increasein travel cost for each mile driven. Matches graph (a).

m � 0.35. 82. The y-intercept is 600 and the slope is whichrepresents the decrease in the value of the wordprocessor each year. Matches graph (d).

�100,

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 11: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.1 Lines in the Plane 11

83. (a)

(b)

00

10

25,000

V � �2300t � 25,000

V � 25,000 � �2300t

V � 25,000 �2000 � 25,000

10 � 0�t � 0�

�0, 25,000�, �10, 2000� (c)

etc.

V � �2300�1� � 25,000 � 22,700t � 1:

V � �2300�0� � 25,000 � 25,000t � 0:

t 0 1 2 3 4 5 6 7 8 9 10

V 25,000 22,700 20,400 18,100 15,800 13,500 11,200 8900 6600 4300 2000

84. (a) Using the points and we have

F �9

5C � 32.

F � 32 �9

5�C � 0�

m �212 � 32

100 � 0�

180

100�

9

5

�100, 212�,�0, 32�

(b)

32.2 � C

58 �95 C

90 �95 C � 32F � 90�:

F � 50

F � 18 � 32

F �95 �10� � 32C � 10�:

�17.8 � C

�32 �95 C

0 �95C � 32F � 0�:

F �95C � 32

F � 350.6

F � 318.6 � 32

F �95 �177� � 32C � 177�:

20 � C

36 �95C

68 �95C � 32F � 68�:

F � 14

F � �18 � 32

F �95 ��10� � 32C � �10�:

C

F 350.6�90�68�50�14�0�

177�32.2�20�10��10��17.8�

85. (a)

(c)

P � 10.25t � 36,500

P � 27t � �16.75t � 36,500�

P � R � C

� 16.75t � 36,500

C � 36,500 � 5.25t � 11.50t (b)

(d)

t � 3561 hours

36,500 � 10.25t

0 � 10.25t � 36,500

R � 27t

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 12: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

12 Chapter 1 Functions and Their Graphs12 Chapter 1 Functions and Their Graphs

86. (a)

(b)

If units.

Algebraically,

(c) If units.

Algebraically, x � �115

�595� �266

3� 49.

x � 49p � 595,

x � �1

15�655� �

2663

� 45.

x � 45,p � 655,

00

1500

100

x ��115

p �2663

x � 50 ��115

�p � 580�

x � 50 �47 � 50

625 � 580� p � 580�

�580, 50�, �625, 47� 87. (a) students per year

(b) 1984: students

1997: students

2000: students

(Answers could vary.)

(c) Let represent 1990.

The slope 341 represents the annual increase in students. It is positive, indicating that Penn State University increased its students from 1991 to 2005.

y � 341t � 75,008

y �477514

�t � 1� � 75,349

y � 75,349 �80,124 � 75,349

15 � 1�t � 1�

�1, 75,349�, �15, 80,124�

t � 0

75,349 � 341�9� � 78,418

75,349 � 341�6� � 77,395

75,349 � 341�7� � 72,962

80,124 � 75,3492005 � 1991

�4775

14� 341

88. Answers will vary. The slope is 341 which is equivalent to the rate of change.

89. False. The slopes are different:

7 � 4

�7 � 0� �

117

4 � 2

�1 � 8�

27

90. False.

The equation of the line joining andis

For

� �18.5

��37

2

� �19.5

y �34

��12� �212

x � �12,

y �34

x �212

.

y � 3 �34

�x � 10�

y � 3 ��9 � 32 � 10

�x � 10�

�2, �9��10, �3�

91.

are the x- and y-intercepts.

−5

−3 9

3

a � 5 and b � �3

�3x � 5y � 15 � 0

x

5�

y

�3� 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 13: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.1 Lines in the Plane 13

92.

and are the x- and -intercepts.

−8

−2

4

6

yba

y �x

3� 2

y � 2�1 �x

6�

x

�6�

y

2� 1

x

a�

y

b� 1 93.

intercepts:

5

−2

−1

2

�4, 0�, �0, �23�

�2x � 12y � �8

�2

3x � 4y �

�8

3

x

4�

y

�23

� 1

94.

Intercepts:

9

3

−1

−3

�12

, 0�, �0, 5�

10x � y � 5

5x �1

2y �

5

2

x12

�y

5� 1 95.

3x � 2y � 6 � 0

x � 6x

2�

y

3� 1 96.

4x � 5y � 20 � 0

x

�5�

y�4

� 1

xa

�yb

� 1

97.

12x � 3y � 2 � 0

�6x �32

y � 1

x

�1�6�

y�2�3

� 1 98.

16x � 15y � 12 � 0

45

x �34

y �35

x

3�4�

y4�5

� 1

xa

�yb

� 1 99. The slope is positive and they-intercept is positive.Matches (a).

100. The slope is negative and they-intercept is negative.Matches (b).

101. Both lines have positiveslope, but their y-interceptsdiffer in sign. Matches (c).

103. No. The line does nothave an x-intercept.

y � 2 105. Yes. Answers will vary.104. No. cannot be writtenin slope-intercept formbecause the slope is undefined.

x � 1

102. The lines intersect in the firstquadrant at a point where Matches (a).x < y.

�x, y�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 14: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

14 Chapter 1 Functions and Their Graphs

■ Given a set or an equation, you should be able to determine if it represents a function.

■ Given a function, you should be able to do the following.

(a) Find the domain.

(b) Evaluate it at specific values.

1. Yes, it does represent a function. Each domainvalue is matched with only one range value.

3. No, it does not represent a function. The domainvalues are each matched with three range values.

7. No, it does not represent a function. The input valuesof 10 and 7 are each matched with two output values.

9. (a) Each element of A is matched with exactly one element of B, so it does represent a function.

(b) The element 1 in A is matched with two elements, and 1 of B, so it does not represent a function.

(c) Each element of A is matched with exactly one element of B, so it does represent a function.

(d) The element 2 of A is not matched to any element of B, so it does not represent a function.

�2

Section 1.2 Functions

107. Yes. x � 20106. Yes. Answers will vary.

117. Answers will vary.

108. Yes. 3x � 10x2 � 1 � �10x2 � 3x � 1

109. No. The term causes the expression to not be a polynomial.x�1 �1x

110. Yes. 2x2 � 2x4 � x3 � 2 � �2x4 � x3 � 2x2 � 2 111. No. This expression is not defined for x � ±3.

112. No. 113. x2 � 6x � 27 � �x � 9��x � 3�

114. x2 � 11x � 28 � �x � 4��x � 7� 115. 2x2 � 11x � 40 � �2x � 5��x � 8�

116. 3x2 � 16x � 5 � �3x � 1��x � 5�

2. No, it is not a function. The domain value of is matched with two output values.

�1

4. Yes, it does represent a function. Every domainvalue is matched with only one range value.

6. No, the table does not represent a function. Theinput values of 0 and 1 are each matched with twodifferent output values.

8. Yes, the table does represent a function. Eachinput value is matched with only one output value.

5. Yes, the relation represents y as a function of x.Each domain value is matched with only onerange value.

Vocabulary Check

1. domain, range, function 2. independent, dependent 3. piecewise-defined

4. implied domain 5. difference quotient

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 15: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.2 Functions 15

11. Each are functions. For each year there correspondsone and only one circulation.

13.

Thus, y is not a function of x. For instance, thevalues both correspond to x � 0.y � 2 and �2

x2 � y2 � 4 ⇒ y � ±�4 � x2

15.

This is a function of x.

y � �x2 � 1

17.

Thus, y is a function of x.

2x � 3y � 4 ⇒ y �13�4 � 2x�

19.

Thus, y is not a function of x. For instance, thevalues and both correspond to x � 2.��3y � �3

y2 � x2 � 1 ⇒ y � ±�x2 � 1

21.

This is a function of x.

y � �4 � x�

23. does not represent as a function of Allvalues of correspond to x � �7.y

x.yx � �7

25.

(a)

(c) f�4t� �1

�4t� � 1�

1

4t � 1

f�4� �1

�4� � 1�

1

5

f�x� �1

x � 1

(b)

(d) f�x � c� �1

�x � c� � 1�

1

x � c � 1

f �0� �1

�0� � 1� 1

27.

(a)

(b)

(c) f�t � 2� � 3�t � 2� � 1 � 3t � 7

f ��4� � 3��4� � 1 � �11

f�2� � 3�2� � 1 � 7

f�t� � 3t � 1

12. million newspapersf �2003� � 7.7

14.

This is not a function of x. For example, the valuesand both correspond to x � 5.y � �2y � 2

y � ±�x � 1

x � y2 � 1

16.

This is a function of x.

y � �x � 5

18.

This is a function of x.

x � �y � 5 ⇒ y � �x � 5.

20.

Thus, y is not a function of x.

x � y2 � 3 ⇒ y � ±�3 � x

22. or

Thus, y is not a function of x.

y � ��4 � x��y� � 4 � x ⇒ y � 4 � x

24. is a function of x, a constant function.y � 8

26.

(a)

(b)

(c)

(d)

� x2 � 2cx � c2 � 2x � 2c

g�x � c� � �x � c�2 � 2�x � c�

g�t � 1� � �t � 1�2 � 2�t � 1� � t2 � 1

g��3� � ��3�2 � 2��3� � 15

g�2� � �2�2 � 2�2� � 0

g�x� � x2 � 2x

10. (a) The element c in A is matched with two elements, 2 and 3 of B, so it is not a function.

(b) Each element of A is matched with exactly one element of B, so it does represent a function.

(c) This is not a function from A to (it represents a function from B to A instead).

(d) Each element in A is matched with exactly one element of B, so it does represent a function.

B

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 16: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

16 Chapter 1 Functions and Their Graphs

35.

(a)

(b)

(c)

is undefined.f �0�

f�t� � �t�t

� �1�1

if t > 0if t < 0

f��3� � ��3��3

� �1

f�3� � �3�3

� 1

f�x� � �x�x

37.

(a)

(b)

(c) f�2� � 2�2� � 2 � 6

f�0� � 2�0� � 2 � 2

f��1� � 2��1� � 1 � �1

f�x� � �2x � 1, x < 0

2x � 2, x ≥ 0

31.

(a)

(b)

(c) f�4x2� � 3 � �4x2 � 3 � 2�x�f �0.25� � 3 � �0.25 � 2.5

f�4� � 3 � �4 � 1

f�y� � 3 � �y

33.

(a)

(b)

(c) q�y � 3� �1

�y � 3�2 � 9�

1

y2 � 6y

q�3� �1

32 � 9 is undefined.

q�0� �1

02 � 9� �

1

9

q�x� �1

x2 � 9

30.

(a)

(b)

(c) V�2r� �4

3��2r�3 �

32�r3

3

V�3

2� �4

3��3

2�3

�4

3�

27

8� �

9�

2

V�3� �4

3��3�3 � 36�

V�r� �4

3�r3

32.

(a)

(b)

(c) f �x � 8� � ��x � 8� � 8 � 2 � �x � 2

f �1� � ��1� � 8 � 2 � 5

f ��8� � ���8� � 8 � 2 � 2

f �x� � �x � 8 � 2

34.

(a)

(b) Division by zero is undefined.

(c) q��x� �2��x�2 � 3

��x�2�

2x2 � 3

x2

q�0� �2�0�2 � 3

�0�2

q�2� �2�2�2 � 3

�2�2�

8 � 3

4�

11

4

q�t� �2t2 � 3

t2

36.

(a)

(b)

(c) f �t� � �t� � 4

f ��4� � ��4� � 4 � 4 � 4 � 8

f �4� � �4� � 4 � 8

f �x� � �x� � 4

29.

(a)

(b)

(c) h�x � 2� � �x � 2�2 � 2�x � 2� � x2 � 2x

h�1.5� � �1.5�2 � 2�1.5� � �0.75

h�2� � 22 � 2�2� � 0

h�t� � t2 � 2t28.

(a)

(b)

(c)

� 7 � 3s � 6 � 1 � 3s

g�s � 2� � 7 � 3�s � 2�

g�73� � 7 � 3�7

3� � 0

g�0� � 7 � 3�0� � 7

g�y� � 7 � 3y

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 17: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.2 Functions 17

43. h�t� �12�t � 3�

41.

(a)

(b)

(c) f�4� � 42 � 1 � 17

f�1� � 4

f��2� � ��2� � 2 � 0

f�x� � �x � 2,4,x2 � 1,

x < 00 ≤ x < 2x ≥ 2

t

1 0 112

12h�t�

�1�2�3�4�5

40.

(a)

(b)

(c) f�1� � 1 � 2�12� � 1 � 2 � �1

f�0� � 02 � 4 � �4

f��2� � ��2�2 � 4 � 4 � 4 � 0

f�x� � �x2 � 4,1 � 2x2

x ≤ 0, x > 0

42.

(a)

(b)

(c) f�1� � 4�1� � 1 � 5

f �12� � 5

f ��2� � 5 � 2��2� � 9

f�x� � �5 � 2x,5,4x � 1,

x < 00 ≤ x < 1x ≥ 1

45. f�x� � ��12x � 4, x ≤ 0

�x � 2�2, x > 0

44.

f�4� � �4 � 2�4 � 2

�2

2� 1

f �5

2� � �52 � 2�52 � 2

1212

� 1

f �3

2� � �3

2 � 2�32 � 2

�12

�12

� �1

f�1� � �1 � 2�1 � 2

�1

�1� �1

f�0� � �0 � 2�0 � 2

�2

�2� �1

f �s� � �s � 2�s � 2

46.

h�5� � �5� � 3 � 2

h�4� � �4� � 3 � 1

h�3� � �3� � 3 � 0

h�2� � 9 � �2�2 � 5

h�1� � 9 � �1�2 � 8

h�x� � �9 � x2,

x � 3,

x < 3

x ≥ 3

x 1 2 3 4 5

8 5 0 1 2h�x�

s 0 1 4

1 1�1�1�1f �s�

52

32

x 0 1 2

5 4 1 092f �x�

�1�2

38.

(a)

(b)

(c) f �1� � 2 � 12 � 1

f �0� � 2�0� � 5 � 5

f ��2� � 2��2� � 5 � 1

f �x� � �2x � 5,

2 � x2,

x ≤ 0

x > 039.

(a)

(b)

(c) f �2� � 2�2�2 � 2 � 10

f �1� � �1�2 � 2 � 3

f ��2� � ��2�2 � 2 � 6

f �x� � �x2 � 2,

2x2 � 2,

x ≤ 1

x > 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 18: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

18 Chapter 1 Functions and Their Graphs

49.

x �43

3x � 4

3x � 4 � 0

f �x� �3x � 4

5� 0

51.

x � �1 or x � 2

�x � 1��x � 2� � 0

x2 � x � 2 � 0

x2 � x � 2

f�x� � g�x�

53.

Since is a polynomial, the domain is all realnumbers x.

f�x�

f�x� � 5x2 � 2x � 1

55.

Domain: All real numbers except t � 0

h�t� �4

t

57.

Domain: all real numbers

f �x� � 3�x � 4

59.

Domain: All real numbers except

x � 0, x � �2

g�x� �1

x�

3

x � 2

61.

Domain: all y > 10.

y > 10

y � 10 > 0

g�y� �y � 2

�y � 10

50.

x �32

2x � 3

2x � 3 � 0

f�x� �2x � 3

7� 0

52.

x � 3 or x � 2

�x � 3��x � 2� � 0

x2 � 5x � 6 � 0

x2 � 2x � 1 � 7x � 5

f�x� � g�x�

54.

Because is a polynomial, the domain is all realnumbers x.

g�x�

g�x� � 1 � 2x2

56.

The domain is all real numbers y � �5.

y � �5

y � 5 � 0

s�y� �3y

y � 5

58.

Domain: or x ≥ 0x ≤ �3

x2 � 3x � x�x � 3� ≥ 0f �x� � 4�x2 � 3x.

60.

The domain is all real numbers and x � 2.x � 0

x�x � 2� � 0

x2 � 2x � 0

h�x� �10

x2 � 2x

62. for numerator, and

for denominator. Domain: x > �6.x � �6

x � 6 ≥ 0f �x� ��x � 66 � x

.

47.

x � 5

3x � 15

f �x� � 15 � 3x � 0 48.

x � �15

5x � �1

f �x� � 5x � 1 � 0

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 19: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.2 Functions 19

63.

Domain:

Range: �0, 2

��2, 2

−4

−6 6

4

f �x� � �4 � x2 64.

Domain: all real numbers

Range: 1 ≤ y

−9

−3

9

9

f �x� � �x2 � 1

65.

Domain:

Range: �0, ��

���, ��

−2

−8 4

6

g�x� � �2x � 3� 66.

Domain: all real numbers

Range: y ≥ 0

−4

−3

14

9

g�x� � �x � 5�

67.

��2, 4�, ��1, 1�, �0, 0�, �1, 1�, �2, 4��

f�x� � x2 68.

��2, 1�, ��1, �2�, �0, �3�, �1, �2�, �2, 1��

f �x� � x2 � 3

69.

��2, 4�, ��1, 3�, �0, 2�, �1, 3�, �2, 4��

f �x� � �x� � 2 70.

��2, 1�, ��1, 0�, �0, 1�, �1, 2�, �2, 3��

f �x� � �x � 1�

71.

�C2

4�A � �� C

2��2

r �C

2�

A � �r2 , C � 2�r

72. in an equilateral triangle and:

A �1

2s �

�3s

2�

�3s2

4

h ��4s2

4�

s2

4�

�3s

2

h ��s2 � �s

2�2

s2 � h2 � �s

2�2

hs

b =s

s2

b � sA �12bh,

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 20: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

20 Chapter 1 Functions and Their Graphs

73. (a) According to the table, the maximum profit is 3375 for .

(b) Yes, P is a function of x.

(c)

P � �30x,45x � 0.15x2,

x ≤ 100x > 100

� 45x � 0.15x2, x > 100

� �105 � 0.15x�x � 60x

� �90 � �x � 100��0.15�x � 60x

� �price per unit��number of units� � �cost��number of units�

Profit � Revenue � Cost

3100100 180

3400

x � 150

74. (a) From the table, the maximum volume seems tobe 1024, corresponding to

(b)

Yes, V is a function of x.

00

7

1200

x � 4.(c)

Domain:

(d)

The function is a good fit. Answers will vary.

0 70

1200

0 < x < 12

� x�24 � 2x�2 � 4x�12 � x�2

� �24 � 2x��24 � 2x�x

V � length � width � height

75.

Since and all lie on the same line, the slopes between any pair of points are equal.

Therefore,

The domain is since A > 0.x > 2,

A �12

xy �12

x� xx � 2� �

x2

2x � 4.

y � 1 �2

2 � x�

xx � 2

1 � y �2

2 � x

1 � y2 � 0

�1 � 02 � x

�x, 0��0, y�, �2, 1�

A �12�base��height� �

12xy.

76.

But so A � 2x�36 � x2, 0 < x < 6.y � �36 � x2,

A � l � w � �2x�y � 2xy

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 21: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.2 Functions 21

79. The domain of is

The domain of is

You can tell by comparing the models to the given data. The models fit the data well onthe domains above.

1 ≤ x ≤ 6.0.505x2 � 1.47x � 6.3

7 ≤ x ≤ 12.�1.97x � 26.3

81.

$4,630 in monthly revenue for November.

f �11� � �1.97�11� � 26.3 � 4.63

83.

corresponds to 1990.t � 0

n�t� � ��6.13t2 � 75.8t � 577,24.9t � 672,

0 ≤ t ≤ 66 < t ≤ 13

t 0 1 2 3 4 5 6 7 8 9 10 11 12 13

Model 577 647 704 749 782 803 811 846 871 896 921 946 971 996

77. (a)

But, or

Thus,

Domain:

(b)

(c) The highest point on the graph occurs atThe dimensions that maximize the

volume are 18 � 18 � 36 inches.x � 18.

00 27

12,000

0 < x < 27

x < 27.

4x < 108

Since y � 108 � 4x > 0

V � �108 � 4x�x2.

y � 108 � 4x.y � 4x � 108,

V � �length��width��height� � yx2 78. (a)

(b)

(c)

P � 5.68x � 98,000

P � 17.98x � �12.30x � 98,000�

Profit � Revenue � Cost

R � 17.98x

Revenue � price per unit � number of units

C � 12.30x � 98,000

Cost � variable costs � fixed costs

80. whichmeans $11, 575 in monthly revenue.f �5� � 0.505�52� � 1.47�5� � 6.3 � 11.575,

82. The values obtained from the model are a close fit for the actual data.

84. (a)

(c)

The maximum occurs at n � 120.

90670

150

720

n ≥ 80 � �12 � 0.05n�n �240n � n2

20,

� �8 � 0.05�n � 80�n

R � �rate��number of people� (b)

The revenue increases, and then decreases.

The maximum revenue occurs when n � 120.

n 90 100 110 120 130 140 150

675 700 715 720 715 700 675R�n�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 22: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

22 Chapter 1 Functions and Their Graphs

85. (a)

(Answers will vary.)

F increases very rapidly as y increases.

(b)

00 50

5,000,000

F�y� � 149.76�10y5�2

y 5 10 20 30 40

4.79 � 1062.33 � 1068.47 � 1051.50 � 1052.65 � 104F�y�

(c) From the table, (slightly above 20). Youcould obtain a better approximation by completingthe table for values of y between 20 and 30.

(d) By graphing together with the horizontal lineyou obtain feet.y 21.37y2 � 1,000,000,

F�y�

y 22 ft

87.

�2cc

� 2, c � 0

f �x � c� � f �x�

c�

2�x � c� � 2xc

f �x� � 2x

89.

�h2 � 3h

h� h � 3, h � 0

�4 � 4h � h2 � 2 � h � 1 � 3

h

f �2 � h� � f �2�

h�

�2 � h�2 � �2 � h� � 1 � 3h

f �x� � x2 � x � 1, f �2� � 3

86. (a) billion dollars

(b) billion dollars/year

This is the average yearly change from 1995 and 2004.

(c)

The model approximates the data well.

(d)

P

4 1560

240

f �2004� � f �1995�2004 � 1995

221 � 72.2

9 16.5

f �2000� 145.6

t 5 6 7 8 9 10 11 12 13 14

72.4 81.6 94.1 109.1 126.1 144.6 163.9 183.5 202.7 221.0P�t�

88.

g�x � h� � g�x�

h�

3hh

� 3, h � 0

g�x � h� � g�x� � �3x � 3h � 1� � �3x � 1� � 3h

g�x � h� � 3�x � h� � 1 � 3x � 3h � 1

g�x� � 3x � 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 23: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.2 Functions 23

91.

�1 � t

t�t � 1� ��1

t, t � 1

f �t� � f �1�t � 1

1t

� 1

t � 1

f �t� �1t, f �1� � 1

93. False. The range of f �x� is ��1, ��.

99. The domain is the set of inputs of the function and the range is the set of corresponding outputs.

92.

��1

2�x � 1�, x � 7f �x� � f �7�

x � 7�

4x � 1

�12

x � 7�

8 � �x � 1�2�x � 1��x � 7� �

7 � x2�x � 1��x � 7�

f�7� �4

7 � 1�

12

f �x� �4

x � 1

94. True. The first number in each ordered pair corresponds to exactly one second number.

100. An advantage of function notation is that it gives a name to the relationship so it can easily bereferenced. When evaluating a function, you see both the input and output values.

102.

�x2 � x � 3

�x � 5��x � 4��x � 1� �3x � 3 � x2 � 4x

�x � 5��x � 4��x � 1�

�3�x � 1�

�x � 5��x � 4��x � 1� �x�x � 4�

�x � 5��x � 1��x � 4�

�3

�x � 5��x � 4� �x

�x � 5��x � 1�3

x2 � x � 20�

xx2 � 4x � 5

101. 12 �4

x � 2�

12�x � 2� � 4x � 2

�12x � 20

x � 2

95. f�x� � �x � 4,4 � x2,

x ≤ 0x > 0

96. f�x� � �1 � x2,x � 1,

x ≤ 0x > 0

97. f�x� � �2 � x,4,x � 1,

x ≤ �2�2 < x < 3x ≥ 3

98. f�x� � �x2,1,5 � x,

x ≤ 11 < x < 4x ≥ 4

90.

f �x � h� � f �x�h

�h�3x2 � 3xh � h2 � 1�

h� 3x2 � 3xh � h2 � 1, h � 0

� h�3x2 � 3xh � h2 � 1�

� 3x2h � 3xh2 � h3 � h

f�x � h� � f�x� � �x3 � 3x2h � 3xh2 � h3 � x � h� � �x3 � x�

� x3 � 3x2h � 3xh2 � h3 � x � hf�x � h� � �x � h�3 � �x � h�

f �x� � x3 � x

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 24: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

24 Chapter 1 Functions and Their Graphs

Section 1.3 Graphs of Functions

■ You should be able to determine the domain and range of a function from its graph.

■ You should be able to use the vertical line test for functions.

■ You should be able to determine when a function is constant, increasing, or decreasing.

■ You should be able to find relative maximum and minimum values of a function.

■ You should know that f is

(a) Odd if

(b) Even if f��x� � f�x�.f��x� � �f�x�.

1. Domain: All real numbers

Range:

f �0� � 1

���, 1]

3. Domain:

Range:

f �0� � 4

�0, 4�

��4, 4�

2. Domain: all real numbers,

Range: all real numbers,

f�0� � 2

���, ��

���, ��

4. Domain: all real numbers,

Range:

f �0� � �3

��3, ��

���, ��

103.

��x � 6��x � 10�

5�x � 3� , x � 0, 12

��2x � 1��x � 6��x � 10�

5�2x � 1��x � 3�

2x3 � 11x2 � 6x5x

�x � 10

2x2 � 5x � 3�

x�2x2 � 11x � 6��x � 10�5x�2x � 1��x � 3�

104. x � 9x � 7

2�x � 9� �x � 7

2�x � 9� �x � 7

2�x � 9� �2�x � 9�

x � 7�

x � 7x � 7

,

Vocabulary Check

1. ordered pairs 2. Vertical Line Test 3. decreasing

4. minimum 5. greatest integer 6. even

5.

Domain: All real numbers

Range: �3, ��

−1

−6 6

7

f �x� � 2x2 � 3 7.

Domain:or

Range: �0, ��

�1, ��x � 1 ≥ 0 ⇒ x ≥ 1

−1

−1 5

3

f �x� � �x � 16.

Domain:

Range: ���, �1�

���, ��

−6

−6

6

2

f �x� � �x2 � 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 25: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.3 Graphs of Functions 25

11.

(a) Domain: all real numbers

(b)

(c) These are the x-intercepts of f.

(d)

(e) This is the y-intercept of f.

(f) The coordinates are

(g) The coordinates are

(h) ��3, f ��3�� � ��3, 6�.f ��3� � ��3�2 � ��3� � 6 � 6.

��1, �4�.f ��1� � ��1�2 � ��1� � 6 � �4.

�1, �6�f �1� � 12 � 1 � 6 � �6.

f �0� � �6

f �x� � x2 � x � 6 � �x � 3��x � 2� � 0 ⇒ x � 3, �2

f �x� � x2 � x � 6

12.

(a) Domain: all real numbers

(b)

(c) These are the x-intercepts of f.

(d)

(e) This is the y-intercept (and x-intercept) of f.

(f) The coordinates are

(g) The coordinates are

(h) ��3, f ��3�� � ��3, �15�.f ��3� � ��3�3 � 4��3� � �27 � 12 � �15.

��1, 3�.f ��1� � �1 � 4��1� � 3.

�1, �3�.f �1� � 1 � 4 � �3.

f �0� � 0

f �x� � x3 � 4x � x�x2 � 4� � x�x � 2��x � 2� � 0 ⇒ x � 0, 2, �2

f �x� � x3 � 4x

13.

(a) Domain: all

(b)

(c) x-intercepts

(d)

(e) y-intercept

(f)

(g)

(h) f ��3� � ��3 � 1� � 2 � 2, ��3, 2�

f ��1� � ��1 � 1� � 2 � 0, ��1, 0�

f �1� � �1 � 1� � 2 � �2, �1, �2�

f �0� � �0 � 1� � 2 � �1

�x � 1� � 2 � 0 ⇒ �x � 1� � 2 ⇒ x � �1, 3

x

f �x� � �x � 1� � 2

9.

Domain: All real numbers

Range: �0, ��

−1

−9 3

7

f�x� � �x � 3�8.

Domain:

Range:

−3

−1

3

3

�0, 2�

��2, 2�

4 � t2 ≥ 0 ⇒ t2 ≤ 4

h�t� � �4 � t2 10.

Domain:

Range: ���, 0�

���, ��

−4

−6

14

6

f �x� � �14 �x � 5�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 26: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

26 Chapter 1 Functions and Their Graphs

14.

(a) Domain: all x

(b) if or

(c) x-intercepts

(d) f �0� � 0 � 4 � 4

x � 2x � �4f �x� � 0

f �x� � �x � 4,4 � x2,

x ≤ 0x > 0

15.

A vertical line intersects the graph just once, so y isa function of x. Graph y1 �

12x2.

y �12x2 16.

y is not a function of x. The vertical lineintersects the graph twice. Graph

and y2 � ��x � 1.y1 � �x � 1

x � 2

x � y2 � 1 ⇒ y � ±�x � 1

17.

A vertical line intersects the graph more than once,so y is not a function of x. Graph the circle as

y2 � ��25 � x2.

y1 � �25 � x2

x2 � y2 � 25 18.

A vertical line intersects the graph just once, so y isa function of x. Solve for y and graph

y �x2 � 1

2x.

x2 � 2xy � 1

19.

f is increasing on ���, ��.

f�x� �3

2x 20.

The graph is decreasing on and increasingon �2, ��.

���, 2�

f �x� � x2 � 4x

21.

f is increasing on

f is decreasing on �0, 2�.

���, 0� and �2, ��.

f�x� � x3 � 3x2 � 2 22.

The graph is decreasing on and increasing on �1, ��.

���, �1�

f �x� � �x2 � 1

23.

(a)

(b) f is constant on ���, ��.

−2

−6 6

6

f �x� � 3 24.

(a)

(b) The graph is increasing on ���, ��.

−6

−4

6

4

f �x� � x

25.

(a)

(b) Increasing on

Decreasing on ���, 0�

�0, ��

−2

−6 6

6

f �x� � x2�3 26.

(a)

(b) The graph is decreasing on �0, ��.

−1

−5

8

1

f �x� � �x3�4

(e) y-intercept

(f)

(g)

(h) f ��3� � �3 � 4 � 1, ��3, 1�

f ��1� � ��1� � 4 � 3, ��1, 3�

f �1� � 4 � 12 � 3, �1, 3�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 27: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.3 Graphs of Functions 27

29.

(a)

(b) Increasing on constant on decreasing on ���, �1�

��1, 1�,�1, ��,

−2

−6 6

6

f �x� � �x � 1� � �x � 1� 30.

(a)

(b) The graph is increasing on constanton and decreasing on ��1, ��.��4, �1�,

���, �4�,

−10

−9

5

1

f �x� � ��x � 4� � �x � 1�

31.

Relative minimum:

−10

−6 12

2

�3, �9�

f�x� � x2 � 6x 32.

Relative minimum:�0.33, �5.33�

−9

−6

9

6

f �x� � 3x2 � 2x � 5 33.

Relative minimum:

Relative maximum:

−8

−6 6

24

��2, 20�

�1, �7�

y � 2x3 � 3x2 � 12x

34.

Relative minimum:

Relative maximum: �0, 15�

�4, �17�

−4

−18

8

18

y � x3 � 6x2 � 15 35.

Relative minimum:

is not a relative maximum because it occurs atthe endpoint of the domain �0, ��.�0, 0�

−1

−1 5

3

�0.33, �0.38�

h�x� � �x � 1��x

36.

Maximum: �2.67, 3.08�

−3

−2

6

4

g�x� � x�4 � x

27.

(a)

(b) Increasing on

Decreasing on ��3, �2�

��2, ��

−3

−9 9

9

f�x� � x�x � 3 28.

(a)

(b) f is decreasing on ���, 1�.

−4

−1

2

3

f �x� � �1 � x

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 28: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

28 Chapter 1 Functions and Their Graphs

37.

(a)

Minimum:

(b)

Minimum:

(c) Answers are the same.

�2, �9�

−18

−12

18

12

�2, �9�

−2−4 2 4 6 8 10 12 14 16

−4

−6

−8

−10

6

8

x

y

f(x) = x2 − 4x − 5

(2, −9)

f �x� � x2 � 4x � 5 38.

(a)

Relative minimum:

(b) Relative minimum:

(c) Answers are the same.

�2, �12�

�2, �12�

f(x) = 3x2 − 12x

(2, −12)

−2−4 2 6 8 10 10

−4

−6

−8

−10

−12

−14

2

4

6

x

y

12 14

f �x� � 3x2 � 12x

39.

(a)

Relative maximum:

Relative minimum:

(b) Relative maximum:

Relative minimum:

(c) Answers are the same.

�1, �2�

��1, 2�

�1, �2�

��1, 2�

f(x) = x3 − 3x(−1, 2)

(1, −2)

−1−3 1 2 3 4 5 6 7

−2

−3

−4

−5

1

3

4

5

x

y

f �x� � x3 � 3x 40.

(a)

Relative maximum:

Relative minimum:

(b) Relative maximum:

Relative minimum:

(c) Answers are the same.

�0, 0�

�2, 4�

�0, 0�

�2, 4�

f(x) = −x3 + 3x2

(0, 0)

(2, 4)

−1−2−3−4−5−6−7 1 2

−2

−3

−4

−5

4

5

x

y

f �x� � �x3 � 3x2

42.

(a)

Relative maximum:

(b) Relative maximum:

(c) Answers are the same.

�1, 4�

�1, 4�

f(x) = 8x − 4x2

(1, 4)

−1−2−3−4−5−6 1 3 4

−2

4

5

2

1

3

x

y

f �x� � 8x � 4x241.

(a)

Relative minimum:

(b) Relative minimum:

(c) Answers are the same.

�1, �2�

�1, �2�

f(x) = 3x2 − 6x + 1

(1, −2)

−1−3−4−5−6−7 −2 1 2 3

−2

−3

1

2

3

x

y

f �x� � 3x2 � 6x � 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 29: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.3 Graphs of Functions 29

43.

−1−3 1 2 3 4 5−1

−2

−3

−4

1

3

4

x

y

f�x� � �2x � 3, x < 0

23 � x, x ≥ 044.

x

y

−2−4−12 4 6

−4

−12

−14

2

4

6

f �x� � �x � 6,2x � 4,

x ≤ �4x > �4

45.

x

y

−1−2−3−4 1 2 3 4−1

−2

−3

1

3

4

5

f �x� � ��x � 4,�4 � x,

x < 0x ≥ 0

46.

x

y

−1 1 2 3 4 5

1

2

3

4

f�x� � �1 � �x � 1�2,�x � 2,

x ≤ 2

x > 247.

x

y

−1−2−4 1 2 3 4−1

−2

−3

1

3

4

5

f �x� � �x � 3,3,2x � 1,

x ≤ 00 < x ≤ 2x > 2

48.

x

y

−1−2−3−4−5 1 2 3 4

−3

−4

−5

1

2

3

4

5

g�x� � �x � 5,�2,5x � 4,

x ≤ �3�3 < x < 1

x ≥ 149.

x

y

−1−2−3−4 2 3 4

−3

−4

−5

1

2

3

f �x� � �2x � 1,x2 � 2,

x ≤ �1x > �1

50.

x

y

−1−2−3−5 1 2 3 4 5

−2−3

1

3

4

5

6

7

h�x� � �3 � x,x2 � 1,

x < 0x ≥ 0

51.

–5 –4 –1 1 2 3 4

–3

–2

2

3

4

5

6

x

y

f �x� � x � 2 52.

1

2

y

–3 –2 –1

–2

–3

–6

–7

541 6x

f �x� � x � 3 53.

–4 –3 1 2 3 4 5

–3

–2

2

1

3

4

5

6

x

y

f �x� � x � 1 � 2

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 30: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

30 Chapter 1 Functions and Their Graphs

54.

–4 –3 –2 –1 2 3 4 5

–4

–5

2

1

3

4

x

y

f �x� � x � 2 � 1 55.

–5 –4 –3 –2 –1 1 2 3 4

–5

–4

1

2

3

4

x

y

f �x� � 2x 56.

–3 –2 –1

2

1

1 2x

y

f �x� � 4x

57.

Domain:

Range:

Sawtooth pattern

�0, 2�

���, ��

−4

−9 9

8

s�x� � 2�14x � 1

4x� 58.

Domain:

Range:

Pattern: Sawtooth

�0, 2�

���, ��

−9

−4

9

8

g�x� � 2�14x � 1

4x�259.

f is neither even nor odd.

� f�t� � �f�t�

� t2 � 2t � 3

f��t� � ��t�2 � 2��t� � 3

63.

The function is odd.

� �f �x�

� �x�1 � x2

f ��x� � ��x��1 � ��x�2 64.

The function is neither even nor odd.

� �f�x�

� f �x�

� �x��x � 5

f��x� � ��x����x� � 5 65.

The function is even.

� g �s�

� 4s2�3

g ��s� � 4 ��s�2�3

60.

f is even.

� x6 � 2x2 � 3 � f�x�.

f��x� � ��x�6 � 2��x�2 � 3 61.

g is odd.

� �g�x�

� �x3 � 5x

g ��x� � ��x�3 � 5��x� 62.

The function is neither odd nor even.

� �h�x�

� h�x�

� �x3 � 5

h��x� � ��x�3 � 5

h�x� � x3 � 5

66. Because the domain is the function is neithereven nor odd.

s ≥ 0, 67.

(a) If f is even, another point is

(b) If f is odd, another point is �32, �4�.�3

2, 4�.��3

2, 4�

68.

(a) If f is even, another point is

(b) If f is odd, another point is �53, 7�.�5

3, �7�.��5

3, �7� 69.

(a) If f is even, another point is

(b) If f is odd, another point is ��4, �9�.

��4, 9�.

�4, 9�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 31: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.3 Graphs of Functions 31

70.

(a) If f is even, another point is

(b) If f is odd, another point is ��5, 1�.

��5, �1�.

�5, �1� 71.

(a) If f is even, another point is

(b) If f is odd, another point is ��x, y�.

��x, �y�.

�x, �y�

72.

(a) If f is even, another point is

(b) If f is odd, another point is ��2a, �2c�.

��2a, 2c�.

�2a, 2c� 73.

−4

−9 9

8

f �x� � 5, even

74.

f is even.

−9

−11

9

1

f �x� � �9 75. is neither even nor odd.

−6

−9 9

6

f �x� � 3x � 2 76. is neither evennor odd.

−6

−1

6

7

f �x� � 5 � 3x

77.

−6

−6 6

2

h �x� � x2 � 4, even 78. is even.

−18

−22

18

2

f �x� � �x2 � 8 79. is neither evennor odd.

−1

−4 2

3

f �x� � �1 � x

80. is neither evennor odd.

−3

−2

6

4

g�t� � 3�t � 1 81. is neither evennor odd.

−1

−5 1

3

f �x� � �x � 2� 82. is neithereven nor odd.

−3

−6

15

6

f �x� � ��x � 5�

83.

���, 4�

4 ≥ x

–1 1 2 3 4 5

1

−1

2

3

4

5

x

yf�x� � 4 � x ≥ 0 84.

��12, ��

x ≥ � 12

4x ≥ �2

4x � 2 ≥ 0

f �x� ≥ 0

–2 –1 1 2

2

3

4

x

yf �x� � 4x � 2

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 32: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

32 Chapter 1 Functions and Their Graphs

85.

�3, �� or ���, �3�

x ≥ 3 or x ≤ �3

x2 ≥ 9–6 –4 2 4 6

–10

–4

–2

2

x

yf�x� � x2 � 9 ≥ 0 86.

���, 0�, �4, ��

x�x � 4� ≥ 0

x2 � 4x ≥ 0

f �x� ≥ 0 1 2 3

–4

–3

–2

–1

x

yf �x� � x2 � 4x

87. (a) The second model is correct. For instance,

� 1.05 � 0.38��12�� � 1.05.

C2�12� � 1.05 � 0.38� ���1

2 � 1���(b)

The cost of an 18-minute 45-second call is

� 1.05 � 0.38�18� � $7.89.

� 1.05 � 0.38���17.75�� � 1.05 � 0.38��18�

C2�184560� � C2�18.75� � 1.05 � 0.38����18.75 � 1���

00 60

25

91.

(a) 1850

1417500

0 ≤ t ≤ 14

P�t� � 0.0108t4 � 0.211t3 � 0.40t2 � 7.9t � 1791

(b) P is increasing from 1990 to 1995

and from 2001 to 2004. P is decreasingfrom 1995 to 2001.

(c) The maximum population was about 1,821,000 in1995 �t � 5.7�.

�t � 11.8��t � 5.7�,�t � 0�

89.

� �x2 � 4x � 3, 1 ≤ x ≤ 3

� ��x2 � 4x � 1� � 2

h � top � bottom 90.

0 ≤ x ≤ 1

� 3 � 4x � x2,

� 3 � �4x � x2�

h � top � bottom

88. Model:

Labels: Total cost

Flat rate

Rate per pound

Equation: C � 9.80 � 2.50x, x > 0

� 2.50x, x > 0

� 9.80

� C

Cos

t of

over

nigh

tde

liver

y(i

n do

llars

)

x

Package weight(in pounds)

87654321

5

10

15

20

25

30

C�Total cost� � �Flat rate� � �Rate per pound�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 33: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.3 Graphs of Functions 33

92. Interval Intake Pipe Drainpipe 1 Drainpipe 2

Open Closed Closed

Open Open Closed

Closed Closed Closed

Closed Closed Open

Open Open Open

Open Closed Open

Open Open Open

Open Open Closed�50, 60�

�45, 50�

�40, 45�

�30, 40�

�20, 30�

�10, 20�

�5, 10�

�0, 5�

93. False. The domain of is the set of allreal numbers.

f �x� � �x2 94. False. The domain must be symmetric about the -axis.y

95. c 96. d 97. b 98. e 100. f99. a

101.

Therefore, is odd.f �x�

� �a2n�1x2n�1 � a2n�1x

2n�1 � . . . � a3x3 � a1x � �f �x�

f��x� � a2n�1��x�2n�1 � a2n�1��x�2n�1 � . . . � a3��x�3 � a1��x�

f�x� � a2n�1x2n�1 � a2n�1x

2n�1 � . . . � a3x3 � a1x

102.

thus, is even.f �x�f ��x� � f �x�;

� a2nx2n � a2n�2 x2n�2 � . . . � a2 x2 � a0 � f �x�

f ��x� � a2n��x�2n � a2n�2��x�2n�2 � . . . � a2��x�2 � a0

f �x� � a2nx2n � a2n�2 x2n�2 � . . . � a2 x2 � a0

103. f is an even function.

(a) is even because

(c) is even becauseg��x� � f ��x� � 2 � f �x� � 2 � g�x�.g�x� � f �x� � 2

g��x� � �f ��x� � �f �x� � g�x�.g�x� � �f �x� (b) is even because

(d) is neither even nor odd becausenor

�g�x�.g��x� � �f ��x � 2� � �f �x � 2� � g�x�g�x� � �f �x � 2�

g��x� � f ����x�� � f �x� � f ��x� � g�x�.g�x� � f ��x�

104. Yes, defines asa function of (But not asa function of )x

yy.xx � y2 � 1 105. No, does not

represent x as a function of y.For instance, and

both lie on the graph.�3, 4���3, 4�

x2 � y2 � 25 106. Answers will vary.

107.

Terms:

Coefficients: �2, 8

�2x2, 8x

�2x2 � 8x 108. Terms:

Coefficient: 3

3x, 10 109.

Terms:

Coefficients:13

, �5, 1

x3

, �5x2, x3

x3

� 5x2 � x3

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 34: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

34 Chapter 1 Functions and Their Graphs

110. Terms:

Coefficient: 7, �2

7x4, �2x2 111. (a)

(b) midpoint � ��2 � 62

, 7 � 3

2 � �2, 5�

� �64 � 16 � �80 � 4�5

d � ��6 � ��2��2 � �3 � 7�2

112. (a)

(b) midpoint � ��5 � 32

, 0 � 6

2 � ��1, 3�

d � ���5 � 3�2 � �0 � 6�2 � �64 � 36 � �100 � 10

113. (a)

(b) midpoint � �52

�32

2,

�1 � 42

� �12

, 32

� �16 � 25 � �41d ����32

�52

2

� �4 � ��1��2

114. (a)

(b) midpoint � ��6 �34

2,

23 �

16

2 � ��218

, 512

d ����6 �34

2

� �23

�16

2

����274 2

� �12

2

��733

4

115.

(a)

(b)

(c) f �x � 3� � 5�x � 3� � 1 � 5x � 16

f��1� � 5��1� � 1 � �6

f�6� � 5�6� � 1 � 29

f �x� � 5x � 1 116.

(a)

(b)

(c)

� �x2 � 3x � 1

� ��x2 � 4x � 4� � x � 2 � 3

f �x � 2� � ��x � 2�2 � �x � 2� � 3

f ��2� � ���2�2 � ��2� � 3 � 1

f �4� � ��4�2 � 4 � 3 � �17

f �x� � �x2 � x � 3

117.

(a)

(b)

(c) f�6� � 6�6 � 3 � 6�3

� 12�3� � 36 � 12�9

f�12� � 12�12 � 3

f�3� � 3�3 � 3 � 0

f �x� � x�x � 3 118.

(a)

(b)

(c) f ��23� � �

12��2

3���23 � 1� �

13�1

3� �19

f �10� � �12�10��10 � 1� � �5�11� � �55

f ��4� � �12��4���4 � 1� � 2�3� � 6

f �x� � �12x�x � 1�

119.

h � 0f�3 � h� � f�3�

h�

�h2 � 4h � 12� � 12h

�h�h � 4�

h� h � 4,

f�3� � 32 � 2�3� � 9 � 12

� h2 � 4h � 12

f �3 � h� � �3 � h�2 � 2�3 � h� � 9 � 9 � 6h � h2 � 6 � 2h � 9

f �x� � x2 � 2x � 9

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 35: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.4 Shifting, Reflecting, and Stretching Graphs 35

120.

�h��h � 6�

h� �h � 6, h � 0

f �h � 6� � f �6�h

���h2 � 6h � 5� � 5

h

f �6� � 5 � 6�6� � 62 � 5

� �h2 � 6h � 5� 5 � 36 � 6h � �36 � 12h � h2� f �6 � h� � 5 � 6�6 � h� � �6 � h�2

f �x� � 5 � 6x � x2

Section 1.4 Shifting, Reflecting, and Stretching Graphs

■ You should know the graphs of the most commonly used functions in algebra, and be able to reproducethem on your graphing utility.

(a) Constant function: (b) Identity function:

(c) Absolute value function: (d) Square root function:

(e) Squaring function: (f ) Cubing function:

■ You should know how the graph of a function is changed by vertical and horizontal shifts.

■ You should know how the graph of a function is changed by reflection.

■ You should know how the graph of a function is changed by nonrigid transformations, like stretches andshrinks.

■ You should know how the graph of a function is changed by a sequence of transformations.

f�x� � x3f�x� � x2

f�x� � �xf�x� � �x�f�x� � xf�x� � c

1.

−4−6 −2

2

2

4

4

6

6

f (x)h (x)

g(x)

x

y 2.

−4

−6

2

4

4

6

6

f(x)h(x)

g(x)

x

y 3.

−4 −2−6 2 4 6

f (x)

h(x)

g(x)

−2

4

x

y

4.

−4

−4

−6

−6

4

4

6

6

f(x)

h (x) g(x)

x

y 5.

−4

−4 −2

−6

−6

4

2

4 6

f(x) h(x)g(x)

x

y 6.

–2–4–6 4 6

–2

2

x

y

f(x)

h (x)

g(x)

Vocabulary Check

1. quadratic function 2. absolute value function 3. rigid transformations

4. 5. 6. (a) ii (b) iv (c) iii (d) ic > 1, 0 < c < 1�f �x�, f ��x�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 36: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

36 Chapter 1 Functions and Their Graphs

10.

–4–6 42 6

–6

6

4

x

y

f(x)

h (x)

g(x)

11.

−2

−2

2

42

4

6

6

f x( )

h x( )

g x( )

x

y 12.

−1−2 1 2 3 4−1

−3

−4

1

2

3

4

f(x)

h(x)

g(x)

x

y

13. (a)

y

x21 3 4 5

4

3

2

1

5

(0, 1)

(1, 2)

(3, 3)

(4, 4)

y

y � f �x� � 2 (b)

y

x1 3 4 5

2

1

−1

−2

−3

(0, 1)

(1, 0)

(3, −1)

(4, −2)

y

y � �f �x� (c)

x2 3 5 61 4

1

2

3

4

−1

−2(2, −1)

(3, 0)

(5, 1)

(6, 2)

y

y � f �x � 2�

(g) Let Then from the graph,

g�8� � f �12�8�� � f �4� � 2

g�6� � f �12�6�� � f �3� � 1

g�2� � f �12�2�� � f �1� � 0

g�0� � f �12�0�� � f �0� � �1

g�x� � f �12x�.

x

y

2 3 4 5 6 7 8−1

−2

−3

−4

1

2

3

4

5

(0, −1)

(2, 0) (6, 1)

(8, 2)

(d)

y

x1 2−13−

2

3

−1

−2

(0, 1)

(1, 2)

(−2, 0)

(−3, −1)

y

y � f �x � 3� (e)

5

4

3

2

1

−2

−3

54321−2 −1−3

(4, 4)

(3, 2)

(0, 2)−

(1, 0)

x

y

y � 2 f �x� (f)

y

x−1−3 −2−4−5

2

3

−2

(0, −1)

(−1, 0)(−3, 1)

(−4, 2)

y

y � f ��x�

7.8

6

4

−2

−4

642−2−4−6

f x( )h x( )

g x( )

x

y 8.6

4

− 2

− 4

−6

64− 4− 6

f x( )

h x( )

g x( )

x

y 9.

−4

−4−6 −2 2

2

4

6

8

6

f(x)

h(x)g(x)

−2

x

y

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 37: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.4 Shifting, Reflecting, and Stretching Graphs 37

14. (a) y

x−1 3−2

1

3

−2

−1

(−2, 3)

(0, 2)

(1, −1)

(3, −2)

y (b) y

x−2 −1 2−3

3

4

2

−1

(−3, 4)

(−1, 3)

(0, 0)

(2, −1)

y (c) y

x−1 2 41

3

4

2

−1

( 1, 4)−

(1, 3)

(2, 0)

(4, 1)−

1

y

(d)2

1

−1

−2

−3

−4

−5

5421−1−2

(5, 1)

(3, 0)

(2, 3)−

(0, 4)−

x

y (e) y

x1−3 −1 2

3

4

1

−1

(2, 4)

(0, 3)

(−1, 0)

(−3, −1)

y (f) y

x−1 1−2

1

3

2

−1

−2

(−2, 2)

( )

( )

(1, 0)

3

1

2

2

0,

3, −

y

(g) Let Then from the graph,

g�32� � f �2�3

2�� � f �3� � �1

g�12� � f �2�1

2�� � f �1� � 0

g�0� � f �2�0�� � f �0� � 3

g��1� � f �2��1�� � f ��2� � 4

2−3 −2 −1

3

4

1

2

−1

(−1, 4)

(0, 3)

, 012( (

, −132( (

x

yg�x� � f �2x�.

15. Horizontal shift three units toleft of (or vertical shift three units upward)

y � x � 3y � x:16. Constant function: y � 7 17. Vertical shift one unit

downward of

y � x2 � 1

y � x2

18. Horizontal shift of y � �x� : y � �x � 2� 19. Reflection in the x-axis and a vertical shift one unitupward of y � �x: y � 1 � �x

20. Reflection in the x-axis and a vertical shift one unitupward of y � 1 � x3y � x3:

21. is reflected in the x-axis,followed by a vertical shift one unit downward.

f �x�y � ��x � 1

22. is shifted verticallyupwards two units.

f �x� � �xy � �x � 2 23. shifted right two units.y � �x � 2 is f �x �

24. is shifted left four units.f�x�y � �x � 4 25. is a vertical stretch of f �x� � �x.y � 2�x

26. is reflected in the y-axis, fol-lowed by a horizontal shift to the right three units.

f �x�y � ��x � 3 27. is shifted left five units.f �x�y � �x � 5�

28. is shifted down three units.f �x� � �x�y � �x� � 3 29. is reflected in the x-axis.f�x�y � ��x�

30. is a reflection in the -axis. In facty � ��x� � �x�.

yy � ��x� 31. is a vertical stretch of f �x�.y � 4�x�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 38: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

38 Chapter 1 Functions and Their Graphs

32. is a vertical shrink.y � �12x� �

12 �x� 33. is obtained from by a reflection

in the x-axis followed by a vertical shift upward offour units.

f�x�g�x� � 4 � x3

34. is obtained by a horizontal shiftof one unit to the right, followed by a reflection inthe -axis.x

g�x� � ��x � 1�3 35. is obtained from by a leftshift of two units and a vertical shrink by a factorof 1

4.

f�x�h�x� �14�x � 2�3

36. is obtained from by aright shift of one unit, a vertical stretch by a factorof two, a reflection in the -axis, and a vertical shiftthree units upward.

x

f�x�h�x� � �2�x � 1�3 � 3 37. is obtained from by ahorizontal stretch followed by a vertical shift two units upward.

f �x�p�x� � �13 x�3

� 2

38. is obtained from by a right shift of two units, followed bya vertical stretch.

f �x�p�x� � �3�x � 2��3

39.

is ahorizontal shift two units to left.

is a vertical shrink.h �x� �12 f �x� �

12�x3 � 3x2�

g�x� � f �x � 2� � �x � 2�3 � 3�x � 2�2

f �x� � x3 � 3x2

40.

is a horizontal shift one unit to the right.

is a horizontal shrink.h�x� � f �3x� � �3x�3 � 3�3x�2 � 2

g�x� � f �x � 1� � �x � 1�3 � 3�x � 1�2 � 2−4

−4

8

4

hf g

f �x� � x3 � 3x2 � 2

41.

reflection in the x-axis and vertical shrink

reflection in the y-axish�x� � f ��x� � ��x�3 � 3��x�2

g�x� � �13 f�x� � �

13�x3 � 3x2�

−4

−6 6

4

fh g

f�x� � x3 � 3x2

42.

is a reflection in the x-axis.

is a horizontal shrink.

−6

−4

6

4

fg h

h�x� � f �2x� � �2x�3 � 3�2x�2 � 2

g�x� � �f �x� � ��x3 � 3x2 � 2�

f �x� � x3 � 3x2 � 2 43. (a)

(b) is obtained from by ahorizontal shift to the left five units, a reflectionin the -axis, and a vertical shift upward twounits.

(c)

(d) g�x� � 2 � f �x � 5�

3

2

1

−2

−3

−4

−5

−6

−7

1−2 −1−4−5−9 −7−8x

y

x

fg�x� � 2 � �x � 5�2

f �x� � x2

−4

−5 7

4

f

hg

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 39: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.4 Shifting, Reflecting, and Stretching Graphs 39

44. (a)

(b) is obtained from f by ahorizontal shift 10 units to the left, a reflectionin the x-axis, and a vertical shift 5 units upward.

(c)

(d) g�x� � �f�x � 10� � 5

642

−4−6−8

−10−12−14

2−2−4−10−16x

y

g�x� � ��x � 10�2 � 5

f �x� � x2 45. (a)

(b) is obtained from by ahorizontal shift four units to the right, a verticalstretch of 2, and a vertical shift upward threeunits.

(c)

(d) g�x� � 3 � 2 f �x � 4�

7

6

5

4

3

2

1

−1 7654321x

y

fg�x� � 3 � 2�x � 4�2

f �x� � x2

46. (a)

(b) is obtained from f by ahorizontal shift two units to the left, a verticalshrink of a reflection in the x-axis, and a vertical shift two units downward.

(c)

(d) g�x� � �14 f�x � 2� � 2

3

2

1

−3

−4

−5

−6

−7

321−1−2−3−4−5−6−7x

y

14,

g�x� � �14�x � 2�2 � 2

f �x� � x2 47. (a)

(b) is obtained from by ahorizontal shift two units to the right followed by a vertical stretch of 3.

(c)

(d) g�x� � 3f �x � 2�

3

2

1

−1

−2

−3

54321−1x

y

fg�x� � 3�x � 2�3

f �x� � x3

48. (a)

(b) is obtained from f bya horizontal shift one unit to the left, a verticalshrink, and a reflection in the x-axis.

(c)

(d) g�x� � �12 f�x � 1�

x

y

−3 −2−4 −1 21 3 4 5

−3

−2

−4

−5

1

2

3

4

5

g�x� � �12 �x � 1�3

f �x� � x3 49. (a)

(b) is obtained from by ahorizontal shift one unit to the right, and avertical shift upward two units.

(c)

(d) g�x� � f �x � 1� � 2

5

4

3

2

1

54321−1−2−3x

y

fg�x� � �x � 1�3 � 2

f �x� � x3

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 40: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

40 Chapter 1 Functions and Their Graphs

50. (a)

(b) is obtained from f bya horizontal shift 3 units to the left, a reflectionin the x-axis, and a vertical shift 10 unitsdownward.

(c)

(d) g�x� � �f�x � 3� � 10

2−2−4−6−8−10−12

2

−4

−6

−8

x

y

g�x� � ��x � 3�3 � 10

f �x� � x3 51. (a)

(b) is obtained from f by a horizontal shift four units to the left, followedby a vertical shift eight units upward.

c)

(d) g�x� � f �x � 4� � 8

x

y

−4−8−12−16 4 8 12−4

4

8

12

16

20

24

g�x� � �x � 4� � 8

f �x� � �x�

52. (a)

(b) is obtained from f by ahorizontal shift three units to the left, followedby a vertical shift nine units upward.

(c)

(d) g�x� � f �x � 3� � 9

x

y

−4−8−12 4 8 12 16−4

4

8

12

16

20

24

28

g�x� � �x � 3� � 9

f �x� � �x� 53. (a)

(b) is obtained from f by ahorizontal shift one unit to the right, a verticalstretch of 2, a reflection in the x-axis, and avertical shift downward four units.

(c)

(d) g�x� � �2f �x � 1� � 4

x

y

−2−4−6−8 2 4 6 8−2

−4

−6

−12

−14

2

g�x� � �2�x � 1� � 4

f �x� � �x�

54. (a)

(b) is obtained from f by ahorizontal shift two units to the right, a verticalshrink, and a vertical shift three units down-ward.

(c)

(d) g�x� �12 f �x � 2� � 3

5

4

3

1

2

−3

−4

−5

54321−1x

y

g�x� �12 �x � 2� � 3

f �x� � �x� 55. (a)

(b) is obtained from f by ahorizontal shift three units to the left, a verticalshrink, a reflection in the x-axis, and a verticalshift one unit downward.

(c)

(d) g�x� � �12 f �x � 3� � 1

5

4

3

1

2

−3

−4

−5

54321−2−4−5x

y

g�x� � �12�x � 3 � 1

f �x� � �x

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 41: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.4 Shifting, Reflecting, and Stretching Graphs 41

56. (a)

(b) is obtained from f bya horizontal shift one unit to the left, a reflec-tion in the x-axis, and a vertical shift six units downward.

(c)

(d) g�x� � �f�x � 1� � 6

108642−2−4−6

2

−4

−8

−10

−12

−14

x

y

g�x� � ��x � 1 � 6

f �x� � �x 57. (a) is a vertical stretch offollowed by a vertical shift of 33.0.

(b)

(c)

corresponds to 1990.

corresponds to 2003.G�0� � F�13�

G��13� � F�0�

�13 ≤ t ≤ 0.

G�t� � F�t � 13� � 33.0 � 6.2�t � 13,

60

130

0

f �t� � �t,F�t� � 33.0 � 6.2�t

58. (a) is a vertical stretch of by 32.3, followed by a vertical shift of 3769.

(b)

(c)

The debt will exceed 10 trillion dollars in 2003.

(d)

corresponds to 2000.

corresponds to 1990.G��10� � M�0�

G�0� � M�10�

�10 ≤ t ≤ 4G�t� � M�t � 10� � 32.3�t � 10�2 � 3769,

t > 13.9

t2 > 192.91

32.3t2 > 6231

M�t� � 32.3t2 � 3769 > 10,000

10,000

140

0

f �t� � t2M�t� � 32.3t2 � 3769

59. False. is a reflection in the y-axis.y � f ��x� 60. False. is a reflection in the x-axis.y � � f �x�

61. (a) is a reflection in the y-axis, so the x-intercepts are and

(b) is a reflection in the x-axis, so the x-intercepts are and

(c) is a vertical stretch, so the x-intercepts are the same: ,

(d) is a vertical shift, so you cannotdetermine the x-intercepts.

(e) is a horizontal shift 3 units to theright, so the x-intercepts are and x � 0.x � 5y � f �x � 3�

y � f �x� � 2

�3.x � 2y � 2 f �x�

x � �3.x � 2y � �f �x�

x � 3.x � �2y � f ��x� 62. (a) is a reflection in the y-axis, so the

x-intercepts are and

(b) is a reflection in the x-axis, so the x-intercepts are the same

(c) is a vertical stretch, so the x-intercepts are the same:

(d) is a vertical shift, so you cannotdetermine the x-intercepts.

(e) is a horizontal shift 2 units to theright, so the x-intercepts are and x � 6.x � 1y � f �x � 2�

y � f �x� � 1

x � �1, 4.y � 2 f �x�

x � �1, 4.y � �f �x�

x � �4.x � 1y � f ��x�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 42: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

42 Chapter 1 Functions and Their Graphs

63. (a) is a reflection in the y-axis, so the graph is increasing on and decreasing on

(b) is a reflection in the x-axis, so the graph is decreasing on and increasingon

(c) is a vertical stretch, so the graph is increasing on and decreasingon

(d) is a vertical shift, so the graph is increasing on and decreasing on

(e) is a horizontal shift one unit tothe left, so the graph is increasing on and decreasing on �1, ��.

���, 1�y � f �x � 1�

�2, ��.���, 2�

y � f �x� � 3

�2, ��.���, 2�

y � 2 f �x�

�2, ��.���, 2�

y � �f �x�

��2, ��.���, �2�

y � f ��x� 64. (a) is a reflection in the y-axis, so the graph is increasing on and decreasing on and

(b) is a reflection in the x-axis, so the graph is increasing on and decreasingon and

(c) is a vertical stretch, so the graph is increasing on and and decreasing on

(d) is a horizontal shift and reflection, so the graph is increasing on and decreasing on and

(e) is a horizontal shift 2 units tothe right, and a vertical shift, so the graph isincreasing on and anddecreasing on �1, 4�.

�4, ��,���, 1�

y � f �x � 2� � 1

�3, ��.���, 0��0, 3�

y � �f �x � 1�

��1, 2�.�2, ��,���, �1�

y �12 f �x�

�2, ��.���, �1���1, 2�

y � �f �x�

�1, ��.�� �, �2���2, 1�

y � f ��x�

65. The vertex is approximately at and the graphopens upward. Matches (c).

�2, 1�

67. The vertex is approximately and the graphopens upward. Matches (c).

�2, �4� 68. The graph of f is shifted to the left approximately four units, reflected in the x-axis,and shifted upward approximately two units.Matches (b).

y � x3

66. The domain is and is approximately on the graph, and Matches (c).

f �x� < 0.�0, �4��0, ���

69. Slope

Slope

Neither parallel nor perpendicular

L2: 9 � 33 � 1

�32

L1: 10 � 22 � 2

� 3 70. Slope

Slope

Neither parallel nor perpendicular

�7 � 5�2 � 1

��12�3

� 4L2:

3 � ��7�4 � ��1� �

105

� 2L1:

71. Domain: All x � 9 72.

Domain: and x � 7x ≥ 5

f �x� ��x � 5x � 7

73. Domain: ⇒ �10 ≤ x ≤ 10100 � x2 ≥ 0 ⇒ x2 ≤ 100

74.

Domain: all real numbers

f �x� � 3�16 � x2

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 43: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.5 Combinations of Functions 43

Section 1.5 Combinations of Functions

■ Given two functions, f and g, you should be able to form the following functions (if defined):

1. Sum:

2. Difference:

3. Product:

4. Quotient:

5. Composition of f with

6. Composition of g with f : �g � f ��x� � g� f�x��g: � f � g��x� � f�g�x��

� f�g��x� � f�x��g�x�, g�x� � 0

� fg��x� � f�x�g�x�� f � g��x� � f�x� � g�x�

� f � g��x� � f�x� � g�x�

1.

x

y

−1−2 1 2 3 4−1

−2

1

2

3

4

h

2.

−1−2−3−4 1 2 3 4−1

−2

−3

−4

1

3

2

4

x

y

h

3.

x

y

−1−2−3 1 2 3 4 5

1

2

4

5

6

7

h

4.

−1−3−4 1 2 3 4−1

−2

−3

−4

1

3

4

x

y

h

5.

(a)

(b)

(c)

(d)

Domain: all x � 3

x � 3� fg��x� �

f �x�g�x� �

x � 3x � 3

,

� fg��x� � f �x�g�x� � �x � 3��x � 3� � x2 � 9

� f � g��x� � f �x� � g�x� � �x � 3� � �x � 3� � 6

� f � g��x� � f �x� � g�x� � �x � 3� � �x � 3� � 2x

g�x� � x � 3 f �x� � x � 3,

Vocabulary Check

1. addition, subtraction, multiplication, division 2. composition

3. 4. inner, outerg�x�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 44: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

44 Chapter 1 Functions and Their Graphs

7.

(a)

(b)

(c)

(d)

Domain: all x � 1.

� f

g� �x� �f �x�g�x�

�x2

1 � x, x � 1

� fg��x� � f�x� � g�x� � x2�1 � x� � x2 � x3

� f � g��x� � f�x� � g�x� � x2 � �1 � x� � x2 � x � 1

� f � g��x� � f�x� � g�x� � x2 � �1 � x� � x2 � x � 1

f �x� � x2, g�x� � 1 � x

8.

(a)

(b)

(c)

(d)

Domain: �� < x < �

� f

g��x� �2x � 5

4�

1

2x �

5

4

� fg��x� � �2x � 5��4� � 8x � 20

� f � g��x� � 2x � 5 � 4 � 2x � 9

� f � g��x� � 2x � 5 � 4 � 2x � 1

f �x� � 2x � 5, g�x� � 4 9.

(a)

(b)

(c)

(d)

Domain: x < 1

� fg��x� �

x2 � 5�1 � x

� fg��x� � �x2 � 5��1 � x

� f � g��x� � x2 � 5 � �1 � x

� f � g��x� � x2 � 5 � �1 � x

f�x� � x2 � 5, g�x� � �1 � x

10.

(a)

(b)

(c)

Domain:

(d)

Domain: and

or x ≤ �2x ≥ 2

x � 0x2 � 4 ≥ 0

��x2 � 1��x2 � 4

x2

� f

g��x� � �x2 � 4 �x2

x2 � 1

x2 ≥ 4 ⇒ x ≥ 2 or x ≤ �2

x2 � 4 ≥ 0

� fg��x� � ��x2 � 4 �� x2

x2 � 1� �x2�x2 � 4

x2 � 1

� f � g��x� � �x2 � 4 �x2

x2 � 1

� f � g��x� � �x2 � 4 �x2

x2 � 1

f�x� � �x2 � 4, g�x� �x2

x2 � 111.

(a)

(b)

(c)

(d)

Domain: x � 0

� fg��x� �

1�x1�x2 � x, x � 0

� fg��x� �1x

�1x2 �

1x3

� f � g��x� �1x

�1x2 �

x � 1x2

� f � g��x� �1x

�1x2 �

x � 1x2

f�x� �1x, g�x� �

1x2

6.

(a)

(b)

(c)

� �2x2 � 7x � 5

� 2x � 2x2 � 5 � 5x

� fg��x� � �2x � 5��1 � x�

� 3x � 6

� 2x � 5 � 1 � x

� f � g��x� � 2x � 5 � �1 � x�

� x � 4 � f � g��x� � 2x � 5 � 1 � x

f �x� � 2x � 5, g�x� � 1 � x

(d)

Domain:

x � 1

1 � x � 0

� f

g��x� �2x � 5

1 � x

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 45: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.5 Combinations of Functions 45

13.

� 8 � 1 � 9

� �32 � 1� � �3 � 2�

� f � g�� 3� � f �3� � g�3� 14.

� 3 � ��4� � 7

� ���2�2 � 1� � ��2 � 2�

� f � g���2� � f ��2� � g��2�

15.

� 1

� �0 � 1� � �0 � 2�

� f � g��0� � f �0� � g�0� 16.

� �1

� �1 � 1� � �1 � 2�

� f � g��1� � f �1� � g�1�

17.

� 30

� 15�2�

� �42 � 1��4 � 2�

� fg��4� � f �4�g�4� 18.

� �280

� 35��8�

� ���6�2 � 1���6 � 2�

� fg���6� � f ��6�g��6�

19.

� �247

�24�7

���5�2 � 1

�5 � 2

� fg���5� �

f ��5�g ��5� 20.

�12

�0 � 10 � 2

� fg��0� �

f �0�g �0� 21.

� 4t2 � 2t � 1

� ��2t�2 � 1� � �2t � 2�

� f � g��2t� � f �2t� � g�2t�

22.

� t2 � 7t � 9

� t2 � 8t � 15 � t � 6

� ��t � 4�2 � 1� � �t � 4 � 2�

� f � g��t � 4� � f �t � 4� � g�t � 4� 23.

� �125t3 � 50t2 � 5t � 2

� �25t2 � 1���5t � 2�

� ���5t�2 � 1���5t � 2�

� fg���5t� � f ��5t�g��5t�

24.

� 27t6 � 18t4 � 3t2 � 2

� �9t4 � 1��3t2 � 2�

� ��3t2�2 � 1��3t2 � 2�

� fg��3t2� � f �3t2�g�3t2� 25.

�1 � t2

t � 2, t � �2 �

t2 � 1�t � 2

���t�2 � 1

�t � 2

� fg���t� �

f ��t�g ��t�

12.

(a)

(b)

(c) � fg��x� �x

x � 1� x3 �

x4

x � 1

� f � g��x� �x

x � 1� x3 �

x � x4 � x3

x � 1

� f � g��x� �x

x � 1� x3 �

x � x4 � x3

x � 1

f �x� �x

x � 1, g�x� � x3

(d)

Domain: x � 0, x � �1

�x

x � 1�

1x3 �

1x2�x � 1�

� fg��x� �

xx � 1

� x3

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 46: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

46 Chapter 1 Functions and Their Graphs

28. 3

9

−3

0f

h g

29. 4

4

−2

−5h

f

g

30. 5

6

−3

−6

f

h g

32.

contributes more to the magnitude of thesum for contributes more to themagnitude of the sum for x > 6.

f �x�0 ≤ x ≤ 2.g�x�

−4

−2

14

f

10

f + g

g

� f � g��x� �x

2� �x

f �x� �x

2, g�x� � �x

33.

contributes more to the magnitude inboth intervals.f �x� � 3x � 2

−6

−9 9

6

f g

f + g

� f � g��x� � 3x � 2 � �x � 5

f �x� � 3x � 2, g �x� � ��x � 5, 34.

g contributes more on both intervals.

−6

−4

6

f

4

f + gg

� �2x2 �32� f � g��x� � �x2 �

12� � ��3x2 � 1�

g�x� � �3x2 � 1,f �x� � x2 �12,

31.

For , contributes more to themagnitude.

For , contributes more to the magnitude.g�x�x > 6

f�x�0 ≤ x ≤ 2

−10

−14 16

10

fg

f + g

f�x� � 3x, g�x� � �x3

10, � f � g��x� � 3x �

x3

10

26.

�t2 � 4t � 3

t, t � 0

��t � 2�2 � 1�t � 2� � 2

� fg��t � 2� �

f �t � 2�g �t � 2�

27. 3

5

−3

−4

h

f

g

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 47: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.5 Combinations of Functions 47

37.

(a)

(b)

(c) � f � g��0� � 20

�g � f ��x� � g � f�x�� � g �3x � 5� � 5 � �3x � 5� � �3x

� f � g��x� � f�g�x�� � f�5 � x� � 3�5 � x� � 5 � 20 � 3x

f�x� � 3x � 5, g�x� � 5 � x

38.

(a)

(b)

(c) is not defined.( f � g��0�

�1

x3� g�x3��g � f ��x� � g�f �x��

�1

x3� �1

x�3

� f �1

x� ( f � g��x� � f �g�x��

f �x� � x3, g�x� �1

x

39. (a) The domain of is or

(b) The domain of is all real numbers.

(c)

The domain of is all real numbers.� f � g�

� f � g��x� � f �g�x�� � f �x2� � �x2 � 4.

g�x� � x2

x ≥ �4.x � 4 ≥ 0f �x� � �x � 4 40. (a) Domain of f :

(b) Domain of g: all real numbers

(c) Domain of

x2

� 3 ≥ 0 ⇒ x ≥ �6

� f � g��x� � f �x2� ��x

2� 3:

x � 3 ≥ 0 ⇒ x ≥ �3

41. (a) The domain of is all realnumbers.

(b) The domain of is all

(c)

The domain of is x ≥ 0.f � g

� ��x �2 � 1 � x � 1, x ≥ 0

� f � g��x� � f �g�x�� � f ��x �x ≥ 0.g�x� � �x

f �x� � x2 � 1 42.

(a) Domain of f :

(b) Domain of g: all x

(c)

Domain: all x

� f � g��x� � f �g�x�� � f �x4� � �x4�1�4 � x

x ≥ 0

g�x� � x4f �x� � x1�4,

35.

(a)

(b)

(c) � f � g��0� � �0 � 1�2 � 1

�g � f ��x� � g � f�x�� � g �x2� � x2 � 1

� f � g��x� � f�g�x�� � f�x � 1� � �x � 1�2

f�x� � x2, g�x� � x � 1 36.

(a)

(b)

(c) ( f � g��0� � 0

� �x � 1� � 1 � x

� � 3�x � 1 �3� 1

� g� 3�x � 1 ��g � f ��x� � g� f �x��

� 3�x3 � x

� 3��x3 � 1� � 1

� f �x3 � 1�

� f � g��x� � f �g�x��

f �x� � 3�x � 1, g�x� � x3 � 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 48: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

48 Chapter 1 Functions and Their Graphs

46.

(a) Domain of f : all

(b) Domain of g: all x

(c)

Domain: all x � 1

� f � g��x� � f �g�x�� � f �x � 1� �2

�x � 1�

x � 0

g�x� � x � 1f �x� �2

�x�, 47. (a) The domain of is all real numbers.

(b) The domain of is all

(c)

Domain: x � ±2

� f � g��x� � f �g�x�� � f � 1

x2 � 4� �1

x2 � 4� 2

x � ±2g�x� �1

x2 � 4

f �x� � x � 2

48. (a) Domain of f : all

(b) Domain of g: all real numbers

(c) Domain of

is all real numbers � 0, �2.

�3

x2 � 2x�

3x�x � 2�

� f � g��x� � f�x � 1� �3

�x � 1�2 � 1

x � ±1 49. (a)

Domain: all x

(b) They are not equal.

0−6 6

8

g ° ff ° g

� x � 4, x ≥ �4

�g � f��x� � g� f�x�� � g��x � 4 � � ��x � 4 �2

� f � g��x� � f�g�x�� � f�x2� � �x2 � 4

50. (a)

Domain: all x

They are equal.

(b)

−6

−4

6

4

f ° g = g ° f

� f � g��x� � �g � f ��x� � x

� �x � 1� � 1 � x

� � 3�x � 13� 1

�g � f ��x� � g� f�x�� � g� 3�x � 1�

� 3�x3 � x � 3��x3 � 1� � 1

� f � g��x� � f�g�x�� � f�x3 � 1� 51. (a)

Domain: all x

The domain of is all real numbers.

(b) They are equal.

f ° g = g ° f−13 7

−6

6

f � g

� 3�13x � 3� � 9 � x

�g � f ��x� � g� f �x�� � g�13x � 3�

� 13�3x � 9� � 3 � x

� f � g��x� � f�g�x�� � f�3x � 9�

43. (a) The domain of is all

(b) The domain of is all real numbers.

(c) The domain of

is all x � �3.

� f � g��x� � f �x � 3� �1

x � 3

g�x� � x � 3

x � 0.f �x� �1x

44. (a) Domain of f : all

(b) Domain of g: all

(c) Domain of

is all x � 0.

x � 0,� f � g��x� � f � 12x� � 2x,

x � 0

x � 0

45. (a) The domain of is all real numbers.

(b) The domain of is all real numbers.

(c)

Domain: all real numbers

� ��3 � x� � 4� � ��x � 1� � �x � 1�� f � g��x� � f �g�x�� � f �3 � x�

g�x� � 3 � x

f �x� � �x � 4�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 49: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.5 Combinations of Functions 49

(c)55. (a)

(b) No, because 24 � 5x � �5x.� f � g��x� � �g � f ��x�

�g � f ��x� � g � f �x�� � g �5x � 4� � 4 � �5x � 4� � �5x

� f � g��x� � f �g�x�� � f �4 � x� � 5�4 � x� � 4 � 24 � 5xx

0 24 0

1 19

2 14

3 9 �15

�10

�5

g� f �x��f �g�x��

56. (a)

(b) They are equal because x � x.

� �x � 1� � 1 � x�g � f ��x� � g�14 �x � 1�� � 4�1

4 �x � 1� � 1

�14 �4x � x� f � g��x� � f�4x � 1� �

14 ��4x � 1� � 1 (c)

(c)

(c)

x

0 0 0

1 1 1

2 2 2

3 3 3

�1�1�1

g� f �x��f �g�x��

57. (a)

(b) No, because �x2 � 1 � x � 1.� f � g��x� � �g � f ��x�

x ≥ �6� x � 1,� �x � 6� � 5

� ��x � 6 �2� 5�g � f ��x� � g � f �x� � � g��x � 6�

� f � g��x� � f �g�x� � � f �x2 � 5� � ��x2 � 5� � 6 � �x2 � 1

58. (a)

(b) They are not equal because x � 6 � 3�x3 � 6.

� 3�x3 � 6�g � f ��x� � g�x3 � 4� � 3��x3 � 4� � 10

� �x � 10� � 4 � x � 6

� f � g��x� � f � 3�x � 10� � � 3�x � 103� 4

x

4

0 6

1 7

2 8

3 9 3�33

3�14

3�7

3�6

3��2�2

g� f �x��f �g�x��

x

0 1 1

3 4�10

�1�5�2

g� f �x��f �g�x��

52. (a)

Domain:

(b) They are equal.

−1

−1

5

3

f ° g = g ° f

x ≥ 0

� f � g��x� � �g � f ��x� � ��x � x1�4 53. (a)

Domain: all x

(b) They are equal.

−1

−3 3

3

f ° g = g ° f

�g � f ��x� � g� f�x�� � g�x2�3� � �x2�3�6 � x4

� f � g��x� � f �g�x�� � f�x6� � �x6�2�3 � x4

54. (a)

Domain: all x

�g � f ��x� � g� f �x�� � g��x�� � ��x�2 � 1

� f � g��x� � f�g�x�� � f��x2 � 1� � ��x2 � 1� (b)

f � g � g � f

−6

−4

6

4

g ° f

f ° g

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 50: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

50 Chapter 1 Functions and Their Graphs

61. (a)

(b) � f

g��2� �f�2�g�2�

�0

2� 0

� f � g��3� � f�3� � g�3� � 2 � 1 � 3 62. (a)

(b) � 4 � 0 � 0� fg��4� � f �4� � g�4�

� 2 � 3 � �1� f � g��1� � f �1� � g�1�

64. (a)

(b) � g�2� � 2�g � f ��3� � g� f �3��

� f �3� � 2� f � g��1� � f �g�1��63. (a)

(b) �g � f��2� � g� f �2�� � g�0� � 4

� f � g��2� � f�g�2�� � f�2� � 0

65. Let This is not a unique solution.For example, if and then as well.� f � g��x� � h�x�g�x� � 2x,f�x� � �x � 1�2

f�x� � x2 and g�x� � 2x � 1, then � f � g��x� � h�x�.

66.

One possibility: Let and

� f � g��x� � f �1 � x� � �1 � x�3 � h�x�

f �x� � x3.g�x� � 1 � x

h�x� � �1 � x�3 67. Let thenThis answer is not unique.

Other possibilities may be:

f�x� � 9�x and g�x� � �x2 � 4�3

f�x� � 3��x and g�x� � 4 � x2 or

f�x� � 3�x � 4 and g�x� � x2 or

� f � g��x� � h�x�.f �x� � 3�x and g�x� � x2 � 4,

68.

One possibility: Let and

� f � g��x� � f �9 � x� � �9 � x � h�x�

f �x� � �x.g�x� � 9 � x

h�x� � �9 � x 69. Let thenAgain, this is not a unique

solution. Other possibilities may be:

or f�x� �1

x � 1 and g�x� � x � 1

f�x� �1

x � 2 and g�x� � x

� f � g��x� � h�x�.f �x� � 1�x and g�x� � x � 2,

(c)59. (a)

(b) No, because 2�x � 1� � 2�x � 3� � 1.� f � g��x� � �g � f ��x�

� 2�x � 3� � 1�g � f ��x� � g � f �x� � � g��x � 3�� � �2x � 2� � 2�x � 1�

� f � g��x� � f �g�x� � � f �2x � 1� � ��2x � 1� � 3� x

0 3

0 2 5

1 4 7

�1

g� f �x��f �g�x��

60. (a)

(b) They are not equal because 6

�3x � 5�

�63x � 5

.

��6

3x � 5�g � f ��x� � g� 6

3x � 5� � �� 63x � 5�

�6

�3x � 5� f � g��x� � f�g�x�� � f��x� �

63��x� � 5

(c)x

0

1 3

2

3 �32�

37

�6�611

�34

65�

65

g� f �x��f �g�x��

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 51: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.5 Combinations of Functions 51

74. (a)

(b) 800

60

0

R1

R2

R3

� 734 � 7.22t � 0.8t2, t � 0, 1, 2, 3, 4, 5, 6

� �480 � 8t � 0.8t2� � �254 � 0.78t�

R3 � R1 � R2

70.

One possibility:

Let and

� f � g��x� � f �5x � 2� �4

�5x � 2�2

f �x� �4

x2 .g�x� � 5x � 2

h�x� �4

�5x � 2�271. Let Then

(Answer is not unique.)� f � g��x� � h�x�.f�x� � x2 � 2x and g�x� � x � 4.

72.

One possibility:

Let and

� �x � 3�3�2 � 4�x � 3�1�2 � h�x�

� f � g��x� � f �g�x�� � f �x � 3�

f �x� � x3�2 � 4x1�2.g�x� � x � 3

h�x� � �x � 3�3�2 � 4�x � 3�1�2 73. (a)

(b)

(c) contributes more to at higher speeds.T�x�B�x�

00 60

300

T

B

R

T�x� � R�x� � B�x� �34x �

115x2

75. corresponds to 1995.t � 5

Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

140 151.4 162.8 174.2 185.6 197 208.4 219.8 231.2 242.6 254

325.8 342.8 364.4 390.6 421.5 457 497.1 541.8 591.2 645.2 703.8

458.8 475.3 497.9 526.5 561.2 602 648.8 701.7 760.7 825.7 896.8y3

y2

y1

76.

represents the total out-of-pocket payments,insurance premiums and other types of premiums in billions of dollars.

yT

2000

150

5

y3

y1

y2

yT

77. gives the area of the circle as a function of time.

� �0.6t�2 � 0.36t2

� A�0.6t�

�A � r��t� � A�r�t��

�A � r��t�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 52: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

52 Chapter 1 Functions and Their Graphs

81. (a)

represents the number of bacteria as afunction of time.

(b)

At time there are 2030 bacteria.

(c) when hours.t 2.3N � 800

t � 6,

�N � T��6� � 10�132� � 20�13� � 600 � 2030

N � T

� 40t2 � 590

� 10�2t � 1�2 � 20�2t � 1� � 600

� N�2t � 1�

�N � T��t� � N�T �t�� 82. (a) Area Hence

(b)

square meters

(c) hours⇒ t 72.2A � 6250 � 27.5625 t

3117

�A � r��36� � 27.5625 �36� � 992.25

�A � r��t� � �5.25�t2� 27.5625 t, t ≥ 0

� r2, r�t� � 5.25�t.

83. represents 3 percent of the amountover $500,000.g� f�x�� � g�x � 500,000� � 0.03�x � 500,000�

84. (a)

(b)

(c)

�S � R��p� � 0.92�p � 1200�

�R � S��p� � 0.92p � 1200

S � 0.92p

R � p � 1200

85. False. but�g � f��x� � g�x � 1� � 6�x � 1�.

� f � g��x� � f�6x� � 6x � 1, 86. True. is only defined if is inthe domain of f.

g�x�� f � g��x� � f �g�x��

(d)

The discount first yields a lower cost.

�S � R��18,400� � 15,824

�R � S��18,400� � 15,728

78. (a)

(b)

(c)

represents the area of the circular base ofthe tank with edge x.A � r

� A�x

2� � �x

2�2

�1

4x2

�A � r��x� � A�r�x��

A�r� � r2

r�x� �x

279.

(a)

represents the cost after t hours.

(b) units

(c)

, or 4 hours 45 minutest � 4.75

30,000

103,0000

x�4� � 50�4� � 200

C �x�t��

� 3000t � 750

� 60�50t� � 750

C�x�t�� � C �50t�

x�t� � 50t

C�x� � 60x � 750

80. 150 miles (450 mph)( hours)

200 miles (450 mph)( hours)

� 50�162t2 � 126t � 25� ��150 � 450t�2 � �200 � 450t�2s � �x2 � y2

t�y �

t�x �

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 53: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.5 Combinations of Functions 53

91.

which shows that g is even.

which shows that h is odd.

� �12 � f�x� � f��x� � �h�x�,

h��x� �12 � f��x� � f����x�� �

12 � f ��x� � f �x�

g��x� �12 � f��x� � f����x�� �

12 � f��x� � f �x� � g�x�,

92. (a)

where g is even and h is odd.

(b)

��1

�x � 1��x � 1� �x

�x � 1��x � 1�

g�x� �12�

1x � 1

�1

�x � 1� �12�

1x � 1

�1

�x � 1�

�12

�2x2 � 2 �12

��4x � �x2 � 1 � ��2x

f �x� �12

��x2 � 2x � 1� � �x2 � 2x � 1� �12

��x2 � 2x � 1� � �x2 � 2x � 1�

h�x��g�x� �

f �x� �12

� f �x� � f ��x� �12

� f �x� � f ��x�

93. �other answers possible��0, �5�, �1, �5�, �2, �7� 94. Three points on the graph of are and ��1, �3.2�.�0, 1�, �1, �2.8�

y �15 x3 � 4x2 � 1

87. Let and be the three siblings, in decreasingage. Then and .

(a)

(b) If then and C � 4.B � 8A � 16,

A � 2B � 2�12C � 6� � C � 12

B �12C � 6A � 2B

CA, B, 88. From Exercise 87, and .

(a) Hence,

(b) If then and A � 14.B � 7C � 2,

C � 2�12A � 6� � A � 12.

2�B � 6� � C and B �12A.

B �12C � 6A � 2B

89. Let and be odd functions, and define Then,

since f and g are both odd

Thus, h is even.

Let and be even functions, and define Then,

since f and g are both even

Thus, h is even.

� h�x�.

� f�x�g�x�

h��x� � f��x�g��x�

h�x� � f �x�g�x�.g�x�f �x�

� f�x�g�x� � h�x�.

� ��f�x���g�x�

h��x� � f��x�g��x�

h�x� � f �x�g�x�.g�x�f �x�

90. The product of an odd function and an even function is odd. Let f be odd and g even. Then

Thus, fg is odd.

� fg���x� � f ��x�g��x� � �f �x�g�x� � �� fg��x�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 54: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

54 Chapter 1 Functions and Their Graphs

Section 1.6 Inverse Functions

■ Two functions f and g are inverses of each other if for every x in the domain of g andfor every x in the domain of f.

■ Be able to find the inverse of a function, if it exists.

1. Replace with y.

2. Interchange x and y.

3. Solve for y. If this equation represents y as a function of x, then you have found If this equationdoes not represent y as a function of x, then f does not have an inverse function.

■ A function f has an inverse function if and only if no horizontal line crosses the graph of f at more thanone point.

■ A function f has an inverse function if and only if f is one-to-one.

f�1�x�.

f�x�

g� f�x�� � xf�g�x�� � x

1.

f�1� f �x�� � f�1�6x� �16�6x� � x

f � f�1�x�� � f �16 x� � 6�1

6 x� � x

f�1�x� �16 x

f �x� � 6x 2.

f�1� f �x�� � f�1�13 x� � 3�1

3 x� � x

f � f�1�x�� � f �3x� �13�3x� � x

f�1�x� � 3x

f �x� �13 x

95.

�other answers possible���24, 0�, ���24, 0�, �0, �24� 96. Three points on the graph of are

and ��1, 14�.�0, 0�, �1, �

14�

y �x

x2 � 5

97.

y � 10x � 38 � 0

y � 2 � 10�x � 4�

y � ��2� �8 � ��2�

�3 � ��4��x � ��4�� 98.

3y � x � 14 � 0

y � 5 �13

�x � 1�

y � 5 �2 � 5

�8 � 1�x � 1�

99.

30x � 11y � 34 � 0

11y � 11 � �30x � 45

y � 1 �5

�11�6�x �32� � �

3011�x �

32�

y � ��1� �4 � ��1�

��1�3� � �3�2��x �32� 100.

2y � x � 2.2 � 0

y � 1.1 � �12

x

y � 1.1 �3.1 � 1.1�4 � 0

�x � 0�

Vocabulary Check

1. inverse, 2. range, domain 3.

4. one-to-one 5. Horizontal

y � x f �1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 55: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 55

9. (a)

(b)

Note that the entries in the tables are the same except that the rows are interchanged.

g� f �x�� � g��72

x � 3� � �2��7

2x � 3� � 67

� ��7x � 6 � 6

7�

7x7

� x

f �g�x� � � f ��2x � 67 � � �

72��

2x � 67 � � 3 �

2x � 62

� 3 � �x � 3� � 3 � x

x 2 0

4 11 18�3�10f �x�

�6�4�2

x 4 11 18

2 0 �6�4�2g�x�

�3�10

5.

f �1� f �x�� � f �1�2x � 1� ��2x � 1� � 1

2�

2x2

� x

f � f �1�x�� � f �x � 12 � � 2�x � 1

2 � � 1 � �x � 1� � 1 � x

f �1�x� �x � 1

2

6.

f � f�1�x�� � f �4x � 1� ��4x � 1� � 1

4�

4x4

� x

f�1�x� � 4x � 1

f �x� �x � 1

47.

f�1� f�x�� � f �1� 3�x � � � 3�x�3� x

f � f�1�x�� � f�x3� � 3�x3 � x

f�1�x� � x3

8.

f �1� f �x�� � f�1�x5� � 5�x5 � x

f � f �1�x�� � f �5�x � � �5�x �5� x

f �1�x� � 5�x

f �x� � x5

3.

f�1� f �x�� � f�1�x � 7� � �x � 7� � 7 � x

f � f�1�x�� � f �x � 7� � �x � 7� � 7 � x

f�1�x� � x � 7

f �x� � x � 7 4.

f�1� f �x�� � f�1�x � 3� � �x � 3� � 3 � x

f � f�1�x�� � f �x � 3� � �x � 3� � 3 � x

f�1�x� � x � 3

f �x� � x � 3

10. (a)

� �x � 9� � 9 � x

g� f �x�� � g�x � 94 � � 4�x � 9

4 � � 9

f �g�x�� � f �4x � 9� ��4x � 9� � 9

4�

4x4

� x (b)

The entries are the same except that the rows are interchanged.

x 1 5 9 13 17

0 1 2�1�2f �x�

x 0 1 2

1 5 9 13 17g�x�

�1�2

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 56: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

56 Chapter 1 Functions and Their Graphs

11. (a)

(b)

Note that the entries in the tables are the same except that the rows are interchanged.

g � f �x� � � g �x3 � 5� � 3��x3 � 5� � 5 � 3�x3 � x

f �g�x� � � f � 3�x � 5� � � 3�x � 5�3� 5 � �x � 5� � 5 � x

x 0 1

4 5 6�3�22f �x�

�1�2�3

x 4 5 6

0 1�1�2�3g�x�

�3�22

12. (a)

g� f �x�� � g�x3

2 � � 3�2�x3

2 � � 3�x3 � x

f �g�x�� � f � 3�2x � �� 3�2x �3

2�

2x2

� x

13. (a)

(b)

Note that the entries in the tables are the same except that the rows are interchanged.

g � f �x�� � g���x � 8� � 8 � ���x � 8�2� 8 � �x � 8� � x

�Since x ≤ 0, �x2 � �x�

x ≤ 0f �g�x� � � f �8 � x2� � ���8 � x2� � 8 � ��x2 � ���x� � x

x 8 9 12 17 24

0 �4�3�2�1f �x�

x 0

8 9 12 17 24g�x�

�4�3�2�1

14. (a)

(b)

The entries in the table are the same except that the rows are interchanged.

g� f �x�� � g� 3�3x � 10� �� 3�3x � 10�3

� 103

��3x � 10� � 10

3�

3x3

� x

f �g�x�� � f �x3 � 103 � � 3�3�x3 � 10

3 � � 10 � 3��x3 � 10� � 10 � 3�x3 � x

x 3 6

0 1 2�1�2f �x�

113

103

23

x 0 1 2

3 6113

103

23g�x�

�1�2

(b)

The entries are the same except that the rows are interchanged.

x 0 1 2

0 412�

12�4f �x�

�1�2

x 0 4

0 1 2�1�2g�x�

12�

12�4

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 57: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 57

Reflections in the line y � x

00

15

10

f

g

18.

g� f�x�� � g�9 � x2� � �9 � �9 � x2� � �x2 � x

f�g�x�� � f ��9 � x � � 9 � ��9 � x �2� 9 � �9 � x� � x

g�x� � �9 � x, x ≤ 9

f�x� � 9 � x2, x ≥ 0

19.

g � f�x�� � g�1 � x3�� 3�1 � �1 � x3� � 3�x3 � x

f�g�x�� � f � 3�1 � x � � 1 � � 3�1 � x ��3� 1 � �1 � x� � x

Reflections in the line y � x

−4

−6 6

4

f

g

20.

g� f �x�� � g� 1

1 � x� �

1 � � 1

1 � x�� 1

1 � x��

1 � x

1 � x�

1

1 � x

1

1 � x

x

1 � x

1

1 � x

�x

1 � x�

x � 1

1� x

f �g�x�� � f �1 � x

x � �1

1 � �1 � x

x ��

1

x

x�

1 � x

x

�1

1

x

� x

f �x� �1

1 � x, x ≥ 0; g�x� �

1 � x

x, 0 < x ≤ 1

Reflections inthe line y � x0

06

4

f

g

21. The inverse is a line through

Matches graph (c).

��1, 0�. 22. The inverse is a line through and

Matches graph (b).

�6, 0�.�0, 6�

15.

Reflections in the line y � x

−4

−6 6

4

f g

g� f�x�� � g�x3� � 3�x3 � x

f�g�x�� � f � 3�x � � � 3�x �3� x 16.

Reflections in the line y � x

−6

−4

6

4

f = g

g� f �x�� � g�1

x� �1

1�x� 1 �

1

x� 1 �

x

1� x

f �g�x�� � f �1

x� �1

1�x� 1 �

1

x� 1 �

x

1� x

f �x� �1

x, g�x� �

1

x

17.

� ��x � 4 �2� 4 � x

g� f�x�� � g��x � 4 � � ��x2 � 4� � 4 � x

f�g�x�� � f�x2 � 4�, x ≥ 0

Reflections in the line y � x

00 15

10

f

g

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 58: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

58 Chapter 1 Functions and Their Graphs

23. The inverse is half a parabola starting at

Matches graph (a).

�1, 0�. 24. The inverse is a reflection in of a third-degreeequation through

Matches graph (d).

�0, 0�.y � x

25.

(a)

Reflection in the line y � x

−4

−6 6

4

fg

g�x� �x2

f �x� � 2x,

(b)

The entries in the tables are the same, except thatthe rows are interchanged.

x 0 1 2

0 2 4�2�4f �x�

�1�2

x 0 2 4

0 1 2�1�2g�x�

�2�4

26.

(a)

The graphs are reflections in the line y � x.

−9 9

−6

6

f

g

g�x� � x � 5

f �x� � x � 5 (b)

The entries in the table are the same except that therows are interchanged.

x 0 3 5

0�2�5�8�10f �x�

�3�5

x 0

0 3 5�3�5g�x�

�2�5�8�10

27.

(a)

Reflection in the line y � x

−10

−12 12

6

f f

gg

g�x� � �5x � 1x � 1

�5x � 11 � x

f �x� �x � 1x � 5,

(b)

The entries in the tables are the same, except thatthe rows are interchanged.

x 0 3 525

14�

15�

12�1f �x�

�1�2

x

0 3 5�1�2g�x�

25

14�

15�

12�1

28.

(a)

Reflection in the line y � x.

−6

−10 11

8

f

f

gg

g�x� �2x � 3x � 1

f �x� �x � 3x � 2

(b)

The entries in the table are the same except that therows are interchanged.

x 0 3 6

0 6 94�

32

16f �x�

�3�4

x 0 6

0 3 6�3�4g�x�

94�

32

16

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 59: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 59

39.

is not one-to-one because somehorizontal lines intersect thegraph twice.

−2

−6 6

6

h

h�x� � �16 � x2

41.

is not one-to-one because thehorizontal line intersectsthe graph at every point on thegraph.

−2

−12 12

14

y � 10f

f �x� � 10 43.

is one-to-one because a horizontal line willintersect the graph at mostonce.

−4

−10 2

4

g

g�x� � �x � 5�3

38.

g does not pass the HorizontalLine Test, so g is not one-to-one.

−6

−2

6

6

g�x� �4 � x

6x240.

is not one-to-one becauseit does not pass theHorizontal Line Test.

−10

−16

10

16

f �x� � �2x�16 � x2

42.

is not one-to-one becauseit does not pass theHorizontal Line Test.

−3

−2

3

2

f �x� � �0.65

44.

is one-to-one becauseit passes the HorizontalLine Test.

−15

−10

15

10

f �x� � x5 � 7 45.

is not one-to-one becausesome horizontal lines intersectthe graph more than once.

−8

−12 12

8

h

h�x� � x � 4 � x � 4 46.

is not one-to-one because itdoes not pass the HorizontalLine Test.

−20

−16

10

4

f �x� � �x � 6x � 6

32. It is the graph of a one-to-onefunction.

33. It is the graph of a one-to-onefunction.

34. It is the graph of a one-to-onefunction.

35.

is one-to-one because a horizontal line will intersect the graph at most once.

−2

−4 8

6

f

f �x� � 3 �12

x 36.

f does not pass the HorizontalLine Test, so f is not one-to-one.

−8

−3

4

5

f �x� �14

�x � 2�2 � 1 37.

is not one-to-one becausesome horizontal lines intersectthe graph twice.

−1

−3 3

3

h

h�x� �x2

x2 � 1

29. Not a function 30. It is the graph of a function,but not one-to-one.

31. It is the graph of a one-to-onefunction.

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 60: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

60 Chapter 1 Functions and Their Graphs

47.

is not one-to-one.

This does not represent y as a function of x. f doesnot have an inverse.

f

y � ± 4�x

x � y4

y � x4

f�x� � x4 48. g is not one-to-one.

For example, g�1� � g��1� � 0.

49.

is one-to-one and has an inverse.f

f �1�x� �5x � 4

3

5x � 4

3� y

5x � 4 � 3y

5x � 3y � 4

x �3y � 4

5

y �3x � 4

5

−9

−6

9

6

f �x� �3x � 4

550.

is one-to-one.

f �1�x� �x � 5

3

x � 5

3� y

x � 5 � 3y

x � 3y � 5

y � 3x � 5

f

f �x� � 3x � 5

51. is not one-to-one, and does not have an

inverse. For example, f �1� � f ��1� � 1.

f �x� �1

x252. is not one-to-one.

For example, h�1� � h��1� � 4.

h�x� �4x2

53.

is one-to-one.

This is a function of x,so f has an inverse.

f�1�x� � �x � 3, x ≥ 0

−3

−3

9

5f

y � �x � 3, x ≥ 0, y ≥ �3

�x � y � 3, y ≥ �3, x ≥ 0

x � � y � 3�2, y ≥ �3, x ≥ 0

y � �x � 3�2, x ≥ �3, y ≥ 0

f �x� � �x � 3�2, x ≥ �3, y ≥ 0 54. is one-to-one.

The inverse is q�1�x� � ��x � 5.

y � ��x � 5

��x � y � 5, y ≤ 5

x � � y � 5�2, y ≤ 5

y � �x � 5�2, x ≤ 5

q�x� � �x � 5�2, x ≤ 5

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 61: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 61

57.

since

x ≥ 0f�1�x� � �x � 2,

y ≤ 2x ≥ 0,y � �x � 2,

x � �y � 2

y � 2 ≤ 0.x � �� y � 2�

y ≤ 2, x ≥ 0x � y � 2, y � x � 2

−9

−6

9

6 f �x� � x � 2, x ≤ 2, y ≥ 0

58.

is not one-to-one.For instance Hence, f does not have an inverse.

f �1� � f ��1�.f

f �x� �x2

x2 � 159.

f�1�x� �x � 3

2

y �x � 3

2

x � 2y � 3

y � 2x � 3

f�x� � 2x � 3

60.

f �1�x� �x

3

x

3� y

x � 3y

y � 3x

f �x� � 3x

Reflections in the line y � x

−6

−4

6

4

ff−1

61.

f�1�x� � 5�x

y � 5�x

x � y5

y � x5

f�x� � x5

Reflections in the line y � x

−4

−6 6

4

ff−1

55.

is one to one.

This is a function of x, so f has an inverse.

−3

−2

9

6

f�1�x� �x2 � 3

2, x ≥ 0

f

y �x2 � 3

2, x ≥ 0, y ≥ �

3

2

x2 � 2y � 3, x ≥ 0, y ≥ �3

2

x � �2y � 3, y ≥ �3

2, x ≥ 0

y � �2x � 3, x ≥ �3

2, y ≥ 0

f�x� � �2x � 3 ⇒ x ≥ �3

2, y ≥ 0 56.

f is one-to-one, so f has an inverse.

f �1�x� � x2 � 2, x ≥ 0

x2 � 2 � y, x ≥ 0, y ≥ 2

x2 � y � 2, x ≥ 0, y ≥ 2

x � �y � 2, y ≥ 2, x ≥ 0

y � �x � 2, x ≥ 2, y ≥ 0

f �x� � �x � 2 ⇒ x ≥ 2, y ≥ 0

Reflections in the line y � x

−2

−4 8

6

ff−1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 62: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

62 Chapter 1 Functions and Their Graphs

65.

f �1�x� � �4 � x2, 0 ≤ x ≤ 2

y � �4 � x2

y2 � 4 � x2

x2 � 4 � y2

x � �4 � y2

y � �4 � x2

f �x� � �4 � x2, 0 ≤ x ≤ 2

62.

f �1�x� � 3�x � 1

3�x � 1 � y

x � 1 � y3

x � y3 � 1

y � x3 � 1

f �x� � x3 � 1

Reflections in the line y � x

−6

−4

6

4

ff−1

63.

f�1�x� � x5�3

y � x5�3

x � y3�5

y � x3�5

f�x� � x3�5

Reflections in the line y � x

−2

−3 3

2

f

f−1

Reflections in the line y � x

00 4

3

f = f−1

64.

f�1�x� � �x

�x � y

x � y2

y � x2

f �x� � x2, x ≥ 0

Reflections in the line y � x

00

6

4

f

f−1

66.

−6

−4

6

4

f

f−1

y � ��16 � x2, 0 ≤ x ≤ 4

y2 � 16 � x2

x2 � 16 � y2

x � �16 � y2, �4 ≤ y ≤ 0

y � �16 � x2

f �x� � �16 � x2, �4 ≤ x ≤ 0 67.

Reflections in the line y � x

f �1�x� �4x

y �4x

xy � 4

x �4y

y �4x

−4

−6 6

4

f = f−1

f �x� �4x

68.

f�1�x� �36x2 , x > 0

y �36x2 , x > 0

x2 �36y

x �6�y

y �6�x

00

15

10

f−1

f

f �x� �6�x

69. If we let then f has an inverse. Note: We could also let

Thus, f�1�x� � �x � 2, x ≥ 0.

�x � 2 � y, x ≥ 0, y ≥ 2

�x � y � 2, x ≥ 0, y ≥ 2

x � �y � 2�2, x ≥ 0, y ≥ 2

y � �x � 2�2, x ≥ 2, y ≥ 0

f �x� � �x � 2�2, x ≥ 2, y ≥ 0

x ≤ 2.��f�x� � �x � 2�2, x ≥ 2,

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 63: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 63

74. Let

Domain Range

Domain Range f �1: y ≥ 4f �1: x ≥ 0

f : y ≥ 0f : x ≥ 4

f �1�x� � �x � 4

y � �x � 4

�x � y � 4

x � � y � 4�2

y � �x � 4�2

f �x� � �x � 4�2, x ≥ 4. 75. Let

Domain Range

Domain Range f �1: y ≥ 0f �1: x ≤ 5

f : y ≤ 5f : x ≥ 0

f �1�x� ��5 � x2

y � ��5 � x��2

y2 �x � 5�2

�5 � x

2

x � 5 � �2y2

x � �2y2 � 5

y � �2x2 � 5

f�x� � �2x2 � 5, x ≥ 0.

76. Let

Domain Range

Domain Range f �1: y ≥ 0f �1: x ≥ �1

f : y ≥ �1f : x ≥ 0

f �1�x� � �2x � 2

2�x � 1� � y2

x �12y2 � 1

y �12x2 � 1

f �x� �12 x2 � 1, x ≥ 0. 77. Let and

because

Domain Range

Domain Range f �1: y ≥ 4f �1: x ≥ 1

f : y ≥ 1f : x ≥ 4

f �1�x� � x � 3, x ≥ 1

y � x � 3

x � y � 3

x ≥ 4. y � x � 3

y � x � 4 � 1

y ≥ 1.f �x� � x � 4 � 1, x ≥ 4

70. If we let then f has aninverse. [Note: We could also let ]

Thus, f �1�x� � 4�1 � x, x ≤ 1.

y � 4�1 � x, x ≤ 1, y ≥ 0

y4 � 1 � x, y ≥ 0, x ≤ 1

x � 1 � y4, y ≥ 0, x ≤ 1

y � 1 � x4, x ≥ 0, y ≤ 1

f �x� � 1 � x4, x ≥ 0 ⇒ y ≤ 1

x ≤ 0.f�x� � 1 � x4, x ≥ 0, 71. If we let then f has an

inverse. Note: We could also let

Thus, f�1�x� � x � 2, x ≥ 0.

x � 2 � y, x ≥ 0, y ≥ �2

x � y � 2, x ≥ 0, y ≥ �2

y � x � 2, x ≥ �2, y ≥ 0

f�x� � x � 2 when x ≥ �2

f�x� � x � 2, x ≥ �2

x ≤ �2.��f�x� � x � 2, x ≥ �2,

72. If we let then f has aninverse. [Note: We could also let ]

when

Thus, f �1�x� � x � 2, x ≥ 0.

x � 2 � y, x ≥ 0, y ≥ 2

x � y � 2, x ≥ 0, y ≥ 2

y � x � 2, x ≥ 2, y ≥ 0

x ≥ 2.f �x� � x � 2

f�x� � x � 2, x ≥ 2

x ≤ 2.f �x� � x � 2, x ≥ 2, 73. Let

Domain Range

Domain Range f �1: y ≥ �3f �1: x ≥ 0

f : y ≥ 0f : x ≥ �3

f �1�x� � �x � 3

y � �x � 3

�x � y � 3

x � � y � 3�2

y � �x � 3�2

f �x� � �x � 3�2, x ≥ �3.

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 64: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

64 Chapter 1 Functions and Their Graphs

78. Let and

because

Domain Range

Domain Range f �1: y ≥ 1f �1: x ≤ �2

f : y ≤ �2f : x ≥ 1

f �1�x� � �x � 1, x ≤ �2

x � 1 � �y

x � �y � 1

y � �x � 1

x ≥ 1. y � �x � 1 � 2 � ��x � 1� � 2

y ≤ �2.f �x� � �x � 1 � 2, x ≥ 1

79.

–4 –3 1 2 3

–3

–2

–1

1

2

3

4

x

y

x

1 2

3 3

�2�1

�4�2

f �x� x

2 1

3 3

�1�2

�2�4

f�1�x�

80.

–3 –2 –1 1 2 3 4 5 6

–3

–2

2

3

4

5

6

x

y

x

4

3

0

6�2

�1

�2

�3

f �x� x

4

3

0

6 �2

�1

�2

�3

f�1�x�

81. because f �12� � 0.f �1�0� �

12 82. because g��2� � 0.g�1�0� � �2

83. � f � g��2� � f �3� � �2 84. g� f ��4�� � g�4� � 6

85. f �1�g�0�� � f �1�2� � 0 86. �g�1� f ��3� � g�1 ��2� � �3

87. �g � f �1 ��2� � g �0� � 2 88. � f �1�g�1 ���2� � f �1��3� � 1

89.

The graph of the inverserelation is an inversefunction since it satisfiesthe Vertical Line Test. −3

−6 6

5

f−1

ff�x� � x3 � x � 1 90. (a) and (b)

(c) Not an inverse function since it does not satisfythe Vertical Line Test.

−3

−2

3

2

h−1h

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 65: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 65

97.

Now find the inverse of

Note: � f � g��1 � g�1� f �1

� f � g��1�x� � 2 3�x � 3

3�8�x � 3� � y

8�x � 3� � y3

x � 3 �18 y3

x �18 y3 � 3

y �18 x3 � 3

� f � g��x� �18 x3 � 3:

� fg��x� � f�g�x�� � f�x3� �18x3 � 3 98.

� 2 3�x � 3

� 3�8�x � 3�

� g�1�8�x � 3��

�g�1� f �1��x� � g�1� f �1�x��

In Exercises 99–102, fx� � x � 4, f�1x� � x � 4, gx� � 2x � 5, g�1x� �x � 5

2.

100.

�x � 3

2

�x � 5 � 8

2

�x � 5

2� 4

� f �1�x � 5

2 � � f �1

� g�1��x� � f �1�g�1�x��99.

�x � 1

2

��x � 4� � 5

2

� g�1�x � 4�

�g�1� f�1��x� � g�1� f�1�x��

91.

The graph of the inverserelation is not an inversefunction since it does notsatisfy the Vertical Line Test.

−4

−6 6

4

g−1

gg�x� �3x2

x2 � 192. (a) and (b)

(c) Inverse function since it satisfies the VerticalLine Test.

−6

−4

6

4

f−1

f

In Exercises 93–98,

93.

94.

95.

96. � 3�3��4 � � 9�4� g�1� 3��4 � �g�1� g�1���4� � g�1�g�1��4��

� f �1� f�1��6� � f�1� f�1 �6�� � f�1�8�6 � 3�� � f�1�72� � 8�72 � 3� � 600

� g�1�0� � 3�0 � 0� g�1�8��3 � 3�� �g�1� f �1���3� � g�1� f �1��3��

� f�1� g�1��1� � f�1�g�1�1�� � f�1� 3�1 � � 8� 3�1 � 3� � 8�1 � 3� � 32

fx� � 18x � 3, f�1x� � 8x � 3�, gx� � x3, g�1x� � 3�x.

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 66: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

66 Chapter 1 Functions and Their Graphs

101. Now find the inverse of

Note that � f � g��1�x� � �g�1� f�1��x�; see Exercise 99.

� f � g��1�x� �x � 1

2

y �x � 1

2

x � 1 � 2y

x � 2y � 1

y � 2x � 1

� f � g��x� � 2x � 1:� f � g��x� � f�g�x�� � f�2x � 5� � �2x � 5� � 4 � 2x � 1.

102.

Note that �g � f ��1 � f �1�g�1.

�g � f ��1�x� �x � 3

2

x � 3

2� y

x � 3 � 2y

x � 2y � 3

y � 2x � 3

� 2x � 3. Now find inverse:� 2x � 8 � 5� 2�x � 4� � 5� g�x � 4� �g � f ��x� � g� f �x��

103. (a) Yes, f is one-to-one. For each European shoesize, there is exactly one U.S. shoe size.

(b)

(c) because

(d)

(e) f �1� f �13�� � f �1 �47� � 13

f � f �1 �41�� � f �8� � 41

f �10� � 43.f �1 �43� � 10

f �11� � 45

104. (a) Yes, g is one-to-one. For each European shoesize, there is exactly one U.S. shoe size.

(b)

(c) because

(d)

(e) g�1�g�5�� � g�1 �37� � 5

g�g�1 �39�� � g�7� � 39

g�9� � 42.g�1 �42� � 9

g�6� � 38

105. (a) Yes, is one-to-one, so exists.

(b) gives the year corresponding to the 10 values in the second column.

(c) because

(d) No, because f �11� � f �15� � 690.4.

f �10� � 650.3.f �1�650.3� � 10

f �1

f �1f

106. (a)

y � number of units produced

x � hourly wage

y � f�1�x� �x � 8

0.75

x � 8

0.75� y

x � 8 � 0.75y

x � 8 � 0.75y

y � 8 � 0.75x (b)

(c) If 10 units are produced, then

(d) If the hourly wage is $22.25, then

y �22.25 � 8

0.75� 19 units.

y � 8 � 0.75�10� � $15.50.

−24

−36 36

24

y

y−1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 67: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.6 Inverse Functions 67

109. We will show that for all x in their domains.Let then Hence,Thus, g�1

� f�1 � � f � g��1. � y � � f � g��1�x�. � g�1�g�y���g�1

� f�1��x� � g�1� f�1�x��f �g� y�� � x ⇒ f�1�x� � g� y�.y � � f � g��1�x� ⇒ � f � g�� y� � x

� f � g��1�x� � �g�1� f�1��x�

110. If is one-to-one, then exists. If is odd, then Consider Then Thus, is odd.f �1f �1��y� � f �1��f �x�� � f �1� f ��x�� � �x � �f �1�y�.

f �x� � y ↔ f �1�y� � x.f ��x� � �f �x�.ff �1f

111. No, the graphs are not reflections of each other in the line y � x.

112. Yes, the graphs are reflections of each other in theline y � x.

113. Yes, the graphs are reflections of each other in theline y � x.

114. Yes, the graphs are reflections of each other in theline y � x.

115. Yes. The inverse would give the time it took tocomplete n miles.

116. Yes, assuming that the population is increasingbetween 1960 and 2005. The inverse would givethe year corresponding to a given population.

107. False. is even, but does not exist.f �1f �x� � x2 108. True. If is the -intercept of then isthe -intercept of f �1.x

�b, 0�f,y�0, b�

117. No. The function oscillates. 118. No, because heights remainconstant, or even decrease,after many years.

119.27x3

3x2 � 9x, x � 0

120.5x2y

xy � 5x�

5x2yx� y � 5� �

5xyy � 5

, x � 0 121.x2 � 366 � x

��x � 6��x � 6�

��x � 6� �x � 6�1

� �x � 6, x � 6

122.x2 � 3x � 40x2 � 3x � 10

��x � 5��x � 8��x � 5��x � 2� �

x � 8x � 2

, x � 5 123.

Yes, y is a function of x.

y � 4x � 3

4x � y � 3

125.

No, y is not a function of x.

y � ±�9 � x2

x2 � y2 � 9124. No. Does not passVertical Line Testx � 5. 126.

Yes, y is a function of x.

y � �x2 � 8

x2 � y � 8

127.

Yes, y is a function of x.

y � �x � 2 128.

No, y is not a function of x.

y � ±�x

y2 � x

x � y2 � 0

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 68: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

68 Chapter 1 Functions and Their Graphs

Section 1.7 Linear Models and Scatter Plots

1. (a)

(b) Yes, the data appears somewhat linear. The moreexperience, corresponds to higher sales, y.x,

Years of experience

Mon

thly

sal

es(i

n th

ousa

nds

of d

olla

rs)

y

x1 2 3 4

10

20

30

40

50

60

2. (a)

(b) No. Quiz scores are dependent on several vari-ables, such as study time, class attendance, etc.

6

6

8

8

10

10

2

2

4

4

12

12

14

14

16

16

x

y

Scor

e on

sec

ond

quiz

Score on first quiz

3. Negative correlation—decreases as increases.x

y 4. No correlation 5. No correlation 6. Positive correlation

7. (a)

(b)

Correlation coefficient: 0.95095

(c)

(d) Yes, the model appears valid.

−4

−1

5

5

y � 0.46x � 1.62

y

x−1−2−4 1 2 3 4

−1

−2

−3

1

2

3

4

5

(−3, 0)(−1, 1)

(0, 2)(2, 3)

(4, 3)

y = x +23

53

8. (a)

(b)

Correlation coefficient:

(c)

(d) The model appears valid.

−7

−2

8

7

�0.94812

y � �1.3x � 2.8

y

x−1−2−3−4 1 3 4

−1

−2

1

2

4

5

6

(−1, 4)

(1, 1)

(2, 1)(0, 2)

(−2, 6)

y = x +32

52

■ You should know how to construct a scatter plot for a set of data

■ You should recognize if a set of data has a positive correlation, negative correlation, or neither.

■ You should be able to fit a line to data using the point-slope formula.

■ You should be able to use the regression feature of a graphing utility to find a linear model for a set of data.

■ You should be able to find and interpret the correlation coefficient of a linear model.

Vocabulary Check

1. positive 2. negative 3. fitting a line to data 4. �1, 1

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 69: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.7 Linear Models and Scatter Plots 69

11. (a)

(b)

(c) or

(d) If cm.d � 0.066�55� � 3.63F � 55,

F � 15.13d � 0.096d � 0.066F

d � 0.07F � 0.3

Force

Elo

ngat

ion

d

F20 40 60 80 100

1

2

3

4

5

6

7

12. (a)

(b)

(c)

Yes, the model is a good fit.

(d) For 2010, and minutes.

For 2015, and minutes.

Yes, the answers seem reasonable.

y � 4.37t � 25

y � 3.76t � 20

8 150

4

y � 0.122t � 1.32

8 150

4

13. (a)

(b)

(c)

Yes, the model is a good fit.

−1 50

1600

y � 136.1t � 836

−1 50

1600

9. (a)

(b)

Correlation coefficient: 0.90978

(c)

(d) Yes, the model appears valid.

−4

−1

8

7

y � 0.95x � 0.92

y

x

(0, 2)

(1, 1)

(2, 2)

(5, 6)

(3, 4)

−1−2 1 2 3 4 5 6

1

2

3

4

5

6

y = x −32

12

10. (a)

(b)

Correlation coefficient:

(c)

(d) The model is somewhat valid.

−5

−1

10

9

�0.95175

y � �1.15x � 6.85

y

x

(4, 3)

(6, 0)

(2, 5)

(0, 7)

(3, 2)

−1 1 2 3 4 5 6 7−1

1

2

3

4

5

6

7y = − x + 7.55

4

(d) For 2005, and or $1,516,500.

For 2010, and or $2,197,000.

Yes, the answers seem reasonable.

(e) The slope is 136.1. It says that the mean salary increases by$136,100 per year.

y � 2197,t � 10

y � 1516.5,t � 5

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 70: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

70 Chapter 1 Functions and Their Graphs

14. (a)

(b)

(c)

Yes, the model is a good fit.

(d) For 2005, and or $46,500.

For 2010, and or $50,700.

Yes, the answers seem reasonable.

y � 50.7,t � 20

y � 46.5,t � 15

8 150

60

y � 0.84t � 33.9

8 150

60

16. (a)

(b)

(c)

The model is a good fit.

(d) For 2050, and or 10,685,000people. Answers will vary.

P � 10,685,t � 50

0 350

15,000

P � 42.0t � 8585

0 350

15,000 17. (a)

(b)

(c)

The model is not a good fit.

(d) For 2050, and or 542,000people. Answers will vary.

P � 542,t � 50

0 350

700

P � 0.6t � 512

0 350

700

18. (a)

Correlation coefficient: 0.81238

(b)

(c) The slope represents the increase in sales due to increased advertising.

(d) For $1500, and or $175,455.y � 175.455x � 1.5

1150

3

250

y � 47.77x � 103.8

15. (a)

(b)

Correlation coefficient: 0.99544

(c)

(d) The model is a good fit.

(e) For 2005,

For 2010,

(f) Answers will vary.

y1 � $46.74.t � 20,

y1 � $38.98.t � 15,

−1 150

60

C � 1.552t � 15.70

−1 150

60

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 71: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Section 1.7 Linear Models and Scatter Plots 71

21. True. To have positive correlation, the y-values tendto increase as x increases.

22. False. The closer to 1 or the better the fit.�1,

23. Answers will vary. 24. Answers will vary.

25.

(a)

(b)

� 2w2 � 5w � 7

f�w � 2� � 2�w � 2�2 � 3�w � 2� � 5

f��1� � 2 � 3 � 5 � 10

f�x� � 2x2 � 3x � 5 26.

(a)

(b)

� 5z2 � 26z � 33

g�z � 2� � 5�z � 2�2 � 6�z � 2� � 1

g��2� � 5�4� � 6��2� � 1 � 33

g�x� � 5x2 � 6x � 1

27.

(a) (b) h�0� � 1 � 0 � 1h�1� � 2�1� � 3 � 5

h�x� � �1 � x2,2x � 3,

x ≤ 0 x > 0

28. (a)

(b) k��1� � ��1�2 � 4 � 5

k��3� � 5 � 2��3� � 11

29.

x � �915 � �

35

15x � �9

6x � 1 � �9x � 8 30.

x � �114

�11 � 4x

3�x � 3� � 7x � 2 31.

x � �14, 32

�4x � 1��2x � 3� � 0

8x2 � 10x � 3 � 0

32.

x �52

, �15

�2x � 5��5x � 1� � 0

10x2 � 23x � 5 � 0 33.

�7 ± �17

4

x �7 ± �49 � 4�4��2�

4

2x2 � 7x � 4 � 0 34.

� 2 ±�62

x �8 ±�64 � 40

4

2x2 � 8x � 5 � 0

19. (a)

Correlation coefficient: 0.79495

(b)

5 180

2000

T � 36.7t � 926 (c) The slope indicated the number of new stores opened per year.

(d)

The number of stores will exceed 1800 near the end of 2013.

t > 23.8

36.7t > 874

T � 36.7t � 926 > 1800

(e)

The model is not a good fit, especially around t � 14.

Year 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Data 1130 1182 1243 1307 1381 1475 1553 1308 1400 1505

Model 1183 1220 1256 1293 1330 1366 1403 1440 1477 1513

20. (a)

(b) The negative slope indicates that the times are decreasing.

(c)

−6 600

8

y � �0.022t � 5.03 (d) The model is not very accurate.

(e) Answers will vary.

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 72: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1

72 Chapter 1 Functions and Their Graphs

1.

−1−4−5 1 2 4 5

−4

−5

2

4

5

x

y 2.

−2−3−4−5−8 1 2

−5

1

4

5

x

y 3.

–4 –2 2 4 6 8

–4

–2

4

6

8

x

(−3, 2) (8, 2)

y

m �2 � 2

8 � ��3� �011

� 0

4. Slope undefined

−2

2

2

4

6

8

8

10

10

12

12

(7, 12)

(7, −1)−2

x

y

�12 � ��1�

7 � 7, 5.

5, 52

2 4 6

−2

2

4

6

, 132(

((

(

x

y

m ��5�2� � 1

5 � �3�2��

3�2

7�2�

3

7

6.

3

2

1

−1

−2

−3

321−1−3 −2

56

34

− , ))

12

52

−, ))

x

y

� �103

�45

� �83

Slope �

56 � ��5

2��

34 �

12

56 �

156

�34 �

24

103

�54

7.

–6 –4 –2 2 4 6

–4

–2

2

6

8

x

(−4.5, 6)

(2.1, 3)

y

m �3 � 6

2.1 � ��4.5��

�3

6.6� �

30

66� �

5

11

��4.5, 6�, �2.1, 3�

8.

−2

−3

−4

−5

−6

−7

1−2−3−4−5−6−7

( 1, 1.2)− −

( 2.7, 6.3)− −

x

y

Slope ��1.2 � 6.3�1 � 2.7

�5.11.7

�5117

� 3 9. (a)

(b) Three additional points:

(other answers possible)

�10 � 4, 1 � 1� � �14, 2�

�6 � 4, 0 � 1� � �10, 1�

�2 � 4, �1 � 1� � �6, 0�

�x � 4y � 6 � 0

4y � 4 � x � 2

y � 1 �14

�x � 2�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 73: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 73

10. (a)

(b) Three additional points:

(other answers possible)

�1 � 2, �1 � 3� � �3, �4�

��1 � 2, 2 � 3� � �1, �1�

��3 � 2, 5 � 3� � ��1, 2�

3x � 2y � 1 � 0

2y � 10 � �3x � 9

y � 5 � �32�x � 3� 11. (a)

(b) Three additional points:

(other answers possible)

�4 � 2, 1 � 3� � �6, 4�

�2 � 2, �2 � 3� � �4, 1�

�0 � 2, �5 � 3� � �2, �2�

�3x � 2y � 10 � 0

2y � 10 � 3x

y � 5 �32�x � 0�

12. (a)

(b) Three additional points:

(other answers possible)

��3 � 3, 4 � 2� � ��6, 6�

�0 � 3, 2 � 2� � ��3, 4�

�3 � 3, 0 � 2� � �0, 2�

2x � 3y � 6 � 0

3y � �2x � 6

y � 0 � �23�x � 3� 13. (a)

(b) Three additional points:

(other answers possible)

�115 � 1, �7 � 1� � �16

5 , �8��6

5 � 1, �6 � 1� � �115 , �7�

�15 � 1, �5 � 1� � �6

5, �6�

5x � 5y � 24 � 0

5y � 25 � �5x � 1

y � 5 � �x �15

y � 5 � �1�x �15�

14. (a)

(b) Three additional points:

(other answers possible)

�10 � 5, �578 � 4� � �15, �89

8 ��5 � 5, �25

8 � 4� � �10, �578 �

�0 � 5, 78 � 4� � �5, �258 �

32x � 40y � 35 � 0

40y � 35 � �32x

y �78 � �

45�x � 0� 15. (a)

(b) Three additional points:

(other answers possible)

�0, 6�, �1, 6�, �2, 6�

y � 6 � 0

y � 6 � 0�x � 2�

16. (a)

(b) Three additional points:

(other answers possible)

�0, 8�, �1, 8�, �2, 8�

y � 8 � 0

y � 8 �horizontal line�

y � 8 � 0 �x � 8� � 0 17. (a) is undefined means that the line is vertical.

(b) Three additional points:

(other answers possible)

�10, 0�, �10, 1�, �10, 2�

x � 10 � 0

m

18. (a) Slope is undefined, line is vertical: or

(b) Three additional points:

(other answers possible)

�5, 0�, �5, 1�, �5, 2�

x � 5 � 0x � 5

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 74: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

74 Chapter 1 Functions and Their Graphs

19.

−3

−3 3

1�Slope � 0�

� 0�x � 2� � 0 ⇒ y � �1

y � 1 ��1 � 14 � 2

�x � 2� 20. Slope is undefined.

Line is vertical.

x � 0−3

−2

3

2

21.

−4

−6 6

4

�27

�x � 1� �27

x �27

⇒ y �27

x �27

y � 0 �2 � 0

6 � ��1� �x � 1� 22.

y � �43

x �223

�3y � 4x � 22

�3y � 18 � 4x � 4

y � 6 �4

�3�x � 1�

−1

−4

11

4

y � 6 �6 � 21 � 4

�x � 1�

23. corresponds to 2008.

Point: , slope:

V � 850t � 5700

V � 12,500 � 850�t � 8�

850�8, 12,500�

t � 8 24.

Point:

V � �115t � 4715

V � 3795 � �115�t � 8�

�8, 3795�

m � �115

25.

Point:

V � 42.70t � 283.90

V � 625.50 � 42.70�t � 8�

�8, 625.50�

m � 42.70 26. corresponds to 2008.

Point: , slope:

V � �5.15t � 114.15

V � 72.95 � �5.15�t � 8�

�5.15�8, 72.95�

t � 8

27.

For the fourth quarter let Then we have

S � 25,000�4� � 110,000 � $210,000.

t � 4.

S � 25,000t � 110,000

S � 160,000 � 25,000�t � 2�

m �185,000 � 160,000

3 � 2� 25,000

�2, 160,000�, �3, 185,000� 28. (a) Point: slope:

(b)

(c) In 2010, and dollars.

(d) when

Algebraically,

t �301.512.75

� 23.6.

V � �12.75t � 301.5 � 0

t � 23.6, �2023�.V � 0

V � 174t � 10

230

240

6

V � �12.75t � 301.5

V � 225 � �12.75�t � 6�

�12.75�6, 225�,

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 75: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 75

29.

(a) Parallel slope:

(b) Perpendicular slope:

−4

−6

8

2

y � �45 x �

25

4x � 5y � 2 � 0

5y � 10 � �4x � 12

y � ��2� � �45�x � 3�

m � �45

y �54x �

234

0 � 5x � 4y � 23

4y � 8 � 5x � 15

y � ��2� �54�x � 3�

m �54

5x � 4y � 8 ⇒ y �54x � 2 and m �

54 30. Slope of given line:

(a)

(b)

240

210

20

30

y �32 x � 15

⇒ 3x � 2y � 30 � 0

y � 3 �32�x � 8� ⇒ 2y � 6 � 3x � 24

y � �23x �

73

⇒ 2x � 3y � 7 � 0

y � 3 � � 23�x � 8� ⇒ 3y � 9 � �2x � 16

m � � 23

31. is a vertical line; the slope is not defined.

(a) Parallel line:

(b) Perpendicular slope:

Perpendicular line:

−7

−3

2

3

� 0 ⇒ y � 2

y � 2 � 0�x � 6�

m � 0

x � �6

x � 4 32. is a horizontal line.

(a) Parallel line through

(b) Perpendicular line through

26

28

18

8

�3,�4�: x � 3

�3, �4�: y � �4

y � 2

33. (a) Not a function. 20 is assigned two differentvalues.

(b) Function

(c) Function

(d) Not a function. No value is assigned to 30.

34. (a) Not a function. u is assigned two different values.

(b) Function

(c) Function

(d) Not a function. is assigned two different values and is unassigned.u

w

35. No, y is not a function of x. Some x-values correspond to two y-values. For example,corresponds to and y � �4.y � 4

x � 136. Yes, y � 2x � 3.

37.

Each x value, corresponds to only one y-value so y is a function of x.

x ≤ 1,

y � �1 � x 38. No, does not pass Vertical Line Test.

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 76: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

76 Chapter 1 Functions and Their Graphs

39.

(a)

(b)

(c)

(d) f �x � 1� � �x � 1�2 � 1 � x2 � 2x � 2

f �b3� � �b3�2 � 1 � b6 � 1

f ��3� � ��3�2 � 1 � 10

f �1� � 12 � 1 � 2

f �x� � x2 � 1 40.

(a)

(b)

(c)

(d) g��x� � ��x�4�3 � x 4�3

g��27� � ��27�4�3 � ��3�4 � 81

g�t � 1� � �t � 1�4�3

g�8� � 84�3 � 24 � 16

g�x� � x4�3

41.

(a)

(b)

(c)

(d) h�2� � 22 � 2 � 6

h�0� � 02 � 2 � 2

h��1� � 2��1� � 1 � �1

h��2� � 2��2� � 1 � �3

h�x� � �2x � 1,x2 � 2,

x ≤ �1x > �1

42.

(a)

(b)

(c)

(d) f �10� �3

2�10� � 5�

315

�15

f �t� �3

2t � 5

f ��2� �3

2��2� � 5�

3�9

� �13

f �1� �3

2�1� � 5� �1

f �x� �3

2x � 5

43. The domain of is all real numbers x � �2.f �x� �x � 1x � 2

44. The domain of is the set of all real numbers.f �x� �x2

x2 � 1

45.

Domain:

Domain: ��5, 5�

�5 � x��5 � x� ≥ 0

25 � x2 ≥ 0

f �x� � �25 � x2 46. The domain of is given by

The domain is ���, �4� � �4, ��.

x2 ≥ 16.

x2 � 16 ≥ 0

f �x� � �x2 � 16

47. The domain of is all real numbers s � 3.g�5� �5s � 53s � 9

48. The domain of is all real numbers � �43

.f �x� �2x � 13x � 4

49. (a)

(b)

� 2.85x � 16,000

� 8.20x � �16,000 � 5.35x�

P�x� � R�x� � C�x�

C�x� � 16,000 � 5.35x

50. in billions of dollarsR�t�

Year 1997 1998 1999 2000 2001 2002 2003 2004

6.744 7.744 8.996 10.5 12.699 11.994 10.448 8.929R�t�

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 77: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 77

51.

� 4x � 2h � 3, h � 0

�4xh � 2h2 � 3h

h

f �x � h� � f �x�

h�

�2x2 � 4xh � 2h2 � 3x � 3h � 1� � �2x2 � 3x � 1�h

� 2x2 � 4xh � 2h2 � 3x � 3h � 1

f �x � h� � 2�x � h�2 � 3�x � h� � 1

f �x� � 2x2 � 3x � 1

52.

� 3x2 � 3xh � h2 � 10x � 5h � 1, h � 0

f �x � h� � f �x�

h�

h�3x2 � 3xh � h2 � 10x � 5h � 1�h

f �x � h� � f �x� � 3x2h � 3xh2 � h3 � 10xh � 5h2 � h

� x3 � 3x2h � 3xh2 � h3 � 5x2 � 10xh � 5h2 � x � h

f �x � h� � �x � h�3 � 5�x � h�2 � �x � h�

53. Domain: All real numbers

Range: y ≤ 3

−4

−6 6

4

54. Domain:

Range: �0, ��−9

−2

9

10

2x2 � 1 ≥ 0 ⇒ x2 ≥12

⇒ ��, ��22 � ��2

2, ��

55. Domain:

Range:

−4

−9 9

80 ≤ y ≤ 6

36 � x2 ≥ 0 ⇒ x2 ≤ 36 ⇒ �6 ≤ x ≤ 6 56. Domain: all real numbers

Range:

−14

−2

4

10�0, ��

57. (a)

(b) is a function of x.y

−6

−9 9

6

y �x2 � 3x

658. (a)

(b) y is a function of x.−14

−8

4

4

y � �23 x � 5

59. (a)

(b) is not a function of x.y

y � ±�2 � 3x

y2 � 2 � 3x

−6

−9 9

6 3x � y2 � 2 60. (a)

(b) y is not a function of x.−12

−8

12

8x2 � y2 � 49

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 78: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

78 Chapter 1 Functions and Their Graphs

61.

(a)

(b) Increasing on and

Decreasing on ��1, 1�

�1, �����, �1�

−6

−9 9

6

f �x� � x3 � 3x 62.

(a)

(b) Increasing on

Decreasing on ���, �3�

�3, ��

−4

−9 9

8

f �x� � �x2 � 9

63.

(a)

(b) Increasing on �6, ��

00 21

14

f �x� � x�x � 6 64.

(a)

(b) Increasing on

Decreasing on ���, �8�

��8, ��

−4

−16 2

8

f �x� � x � 8 2

65.

Relative minima: and

Relative maximum:

−18

−4

18

20�0, 16�

�2, 0���2, 0�

f �x� � �x2 � 4�2 66.

Relative minimum: �0.5, �1.25�

−4.5

−3

4.5

3

f �x� � x2 � x � 1

67.

Relative maximum: �3, 27�

−10

−10

10

30

h �x� � 4x3 � x4 68.

Relative maximum:

Relative minimum: �2.67, �10.48�

�0, �1�

−12

−12

12

4

f �x� � x3 � 4x2 � 1

69.

1

56

1 2 4 5 6−1−3−4−5−6−2−3−4−5−6

x

y

f �x� � �3x � 5,x � 4,

x < 0 x ≥ 0

70.

654321−1−2−3−4−5

1110

654321

x

y

f �x� � �x2 � 7,x2 � 5x � 6,

x < 1 x ≥ 1

71.

–5 –1–2 1 2 3 4

–3

–2

3

4

5

6

x

y

f �x� � �x� � 3

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 79: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 79

72.

–5–6 –1–4 1 2 3

–4

–3

–2

2

3

4

5

x

y

f �x� � �x � 2� 73.

Even

� f �x�

� x2 � 6

f ��x� � ��x�2 � 6

74.

and

Neither even nor odd

f ��x� � �f �x�

� f �x�

� x2 � x � 1

f ��x� � ��x�2 � ��x� � 1

78.

Even

f ��x� � 3��x�2�5 � 3x2�5 � f �x� 79. f �x� � �2 is a constant function.

75.

f is even.

� f �x�

� �x2 � 8�2

f ��x� � ���x�2 � 8�2

76. is neither even nor odd.f �x� � 2x3 � x2 77. and

Neither even nor odd

(Note that the domain of is x ≥ 0.)f

f ��x� � �f �x�f ��x� � 3��x�5�2 � f �x�

80. is the parent function. is obtained from by a reflection in the axis, followed by a verticalshift five units upward.

f �x� � �x � 5 � �g�x� � 5

x-gfg�x� � x 81. is the parent function. is obtained

from by a horizontal shift two units to the right,followed by a vertical shift one unit upward.

f �x� � �x � 2�2 � 1 � g�x � 2� � 1

gfg�x� � x2

82. is obtained from by areflection in the axis, followed by a vertical shifttwo units downward.

g�x� � �f �x� � 2

x-f �x� � x3g�x� � �x3 � 2 83. is obtained from by a

vertical shift three units upward.

g�x� � f �x� � 3

f �x� � x g�x� � x � 3

84. is obtained from by ahorizontal shift three units to the right followed bya reflection in the x-axis.

g�x� � �f �x � 3�

f �x� � �xg�x� � ��x � 3 85.

is a reflection in the y-axis.y � f ��x�

−2−2−4−8−10 2

4

6

8

4 6

−6

−8

−4

x

y

(−8, −4)

(−4, 2) (1, 2)

(4, −4)

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 80: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

80 Chapter 1 Functions and Their Graphs

86.

is a reflection inthe x-axis.y � � f �x�

−2−4−6 2

4

2

6

8

6 8 10

−6

−8

−4

x

y

(−4, 4) (8, 4)

(4, −2)(−1, −2)

87.

is a vertical shifttwo units downward.y � f �x� � 2

−4−6

4

2

6 8 10

−6

−8

−10

−12

−4

x

y

(−4, −6)

(−1, 0) (4, 0)

(8, −6)

88.

is a horizontalshift one unit to the right.y � f �x � 1�

−4−6

4

6

8

2

42 8 10

−6

−8

−4

x

y

(−3, −4)

(0, 2) (5, 2)

(9, −4)

89. (a)

(b) is a vertical shift six units downward.

(c)

(d) h�x� � f �x� � 6

1

2

3

−2

−3

−7

−1−3−4−5 1 3 4 5x

y

h

f �x� � x2 90. (a)

(b) is a reflection in the axis, followed by avertical shift three units downward.

(c)

(d) h�x� � �f �x� � 3

−1−2−3−4−5 1 2 3 4 5

1

2

−2

−3

−6

−7

−8

x

y

x-h

f �x� � x2

92.

(a)

(b) The graph of h is a horizontal shift of f twounits to the left, followed by a reflection in the x-axis, followed by a vertical shift eight unitsdownward.

(c)

(d) � �f �x � 2� � 8h �x� � ��x � 2�2 � 8

–1

–20–18–16–14–12

–6–4

–2–3–4–5–6–7–8–9 1x

y

f �x� � x2

h �x� � ��x � 2�2 � 8

93. (a)

(b) is a horizontal shift two units to the right, a reflection in the axis, followed by a vertical shift eight units downward.

(d) h�x� � �f �x � 2� � 8

x-h

f �x� � x2

91.

(a)

(b) The graph of h is a horizontal shift of f twounits to the right, followed by a vertical shiftfive units upward.

(c)

(d) � f �x � 2� � 5h �x� � �x � 2�3 � 5

–2 –1

–2–3

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

x

y

f �x� � x3

h �x� � �x � 2�3 � 5

(c)

−2−4−6−8 2

2

4 6 8

−6

−8

−10

−12

−14

−4

−2

x

y

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 81: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 81

94.

(a)

(b) The graph of h is a horizontal shift of f threeunits to the right, followed by a vertical shrinkof followed by a vertical shift six units downward.

(c)

(d)

� 12 f �x � 3� � 6

h�x� �12�x � 3�2 � 6

y

–1

–2

–3

–4

–5

–6

–7

–8

–2 54321 7 8x

12,

f �x� � x2

h�x� �12�x � 3�2 � 6 95.

(a)

(b) The graph of h is a reflection of f in the x-axis,followed by a vertical shift five units upward.

(c)

(d)

� �f �x� � 5

h�x� � ��x � 5

1

32

4

78

65

y

–1

–2

54321 6 7 8 9x

f �x� � �x

h�x� � ��x � 5

96.

(a)

(b) The graph of h is a vertical stretch of f of 2,followed by a vertical shift five units upward.

(c)

(d)

� 2 f �x� � 5

h�x� � 2�x � 5

x

y

−2 2 4 6 8 10 12−2

2

4

6

8

10

12

f �x� � �x

h�x� � 2�x � 5 97.

(a)

(b) The graph of is a horizontal shift of one unitto the right, followed by a vertical shift threeunits upward.

(c)

(d) h�x� � f �x � 1� � 3

−1−2−3−4 1

1

2

3

4

5

6

2 3 4−1

−2

x

y

h

f �x� � �x

h�x� � �x � 1 � 3

98.

(a)

(b) The graph of h is a vertical shift of f nine unitsupward.

(c)

(d)

� f �x� � 9

h�x� � x � 9

2

4

6

16

14

12

8

y

–4–5 –3 –2 –1 4321 5x

f �x� � x h�x� � x � 9 99.

(a)

(b) is a vertical shrink, followed by a reflectionin the axis, followed by a vertical shift nineunits upward.

(c)

(d) h�x� � �12 f �x� � 9

−2−4−6−8 2

2

4

6

10

12

4 6 8

−4

−2

x

y

x-h

f �x� � x h�x� � �

12 x � 9

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 82: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

82 Chapter 1 Functions and Their Graphs

100.

(a)

(b) is a horizontal shift eight units to the left, followed by a vertical shift one unit downward.

(c) (d) h�x� � f �x � 8� � 1

−2−4−6−10−12 2

2

4

8

−4

−6

−8

−2

x

y

h

f �x� � x h�x� � x � 8 � 1

101.

� �7

� �5 � 2

� �3 � 2�4�� � �4

� f � g��4� � f�4� � g�4� 102.

� 70

� �7 � 77

� f � h��5� � f�5� � h�5� 103.

� �42

� �47 � 5

� f � g��25� � f �25� � g�25�

104. �g � h��1� � g�1� � h�1� � 1 � 5 � �4 105.

� �1��5� � 5

� fh��1� � f�1�h�1� � �3 � 2�1���3�1�2 � 2�

106. g

h��1� �g�1�h�1�

�1

5107.

� 23

� 3��7 �2� 2

� h��7 ��h � g��7� � h�g�7�� 108. �g � f���2� � g�7� � �7

109.

� �97

� f �50�

� f � h���4� � f �h��4�� 110.

� �110

� g�110�

�g � h��6� � g�h�6�� 111.

� �x � 3�2 � h�x�

� f � g��x� � f �x � 3�

f �x� � x2, g�x� � x � 3

112.

� f � g��x� � f �1 � 2x� � �1 � 2x�3 � h�x�

f �x� � x3, g�x� � 1 � 2x 113.

� f � g��x� � f �4x � 2� � �4x � 2 � h�x�

f �x� � �x, g�x� � 4x � 2

114.

� f � g��x� � f ��x � 2�2� � 3��x � 2�2 � h�x�

f �x� � 3�x, g�x� � �x � 2�2 115.

� f � g��x� � f �x � 2� �4

x � 2� h�x�

f �x� �4x, g�x� � x � 2

116.

� f � g��x� � f �3x � 1� �6

�3x � 1�3 � h�x�

f �x� �6x3, g�x� � 3x � 1 117. 3

140

0

y1

y2

y1 + y2

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 83: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 83

118.

For 2008, let

or about 2,987,000 students.� y1 � y2��18� � 2.987,

t � 18.

y1 � y2 � �0.00204t2 � 0.0015t � 1.021� � �0.0274t � 0.785�

119.

f �1� f �x�� � f �1�6x� �16�6x� � x

f � f �1�x�� � f �16 x� � 6�1

6 x� � x

f �1�x� �16 x

f �x� � 6x 120.

f �1� f �x�� � f �1�x � 5� � �x � 5� � 5 � x

f � f �1�x�� � f �x � 5� � �x � 5� � 5 � x

f �1�x� � x � 5

f �x� � x � 5

121.

� 212

x � 3 � 3� � 212

x� � x

f �1� f �x�� � f �112

x � 3�

�12

�2�x � 3�� � 3 � x � 3 � 3 � x

f � f �1�x�� � f �2�x � 3��

f �x� �12

x � 3 ⇒ f �1�x� � 2�x � 3� � 2x � 6 122.

� 5x � 45 � � 4 � x � 4 � 4 � x

f �1� f �x�� � f �1x � 45 �

f � f �1�x�� � f �5x � 4� �5x � 4 � 4

5�

5x5

� x

f �x� �x � 4

5 ⇒ f �1�x� � 5x � 4

123. (a)

Reflection in the line y � x

−6

−9 9

6

g

f

(b)

The entries in the table are the same exceptthat their rows are interchanged.

x 23 7 3

0 1 3�1�5g�x�

�9�1

x 0 1 3

23 7 3 �9�1f �x�

�1�5

124.

(a)

Reflections in y � x

00

9

6

g

f

f �x� � �x � 1, g�x� � x2 � 1, x ≥ 0

(b)

The entries are the same, except that the rowsare interchanged.

x 0 3 8 15

0 1 2 3 4f �x�

�1

x 0 1 2 3 4

0 3 8 15�1g�x�

125.

passes the Horizontal Line Test,and hence is one-to-one and has an inverse� f�1�x� � 2�x � 3��.

f �x� �12 x � 3

−6

−9 9

6 126.

does not pass the Horizontal LineTest. Not one-to-onef �x� � �x � 1�2

−6

−2

6

6

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 84: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

84 Chapter 1 Functions and Their Graphs

127.

passes the Horizontal Line Test, and

hence is one-to-one.

h�t� �2

t � 3

−6

−9 9

6 128.

passes the Horizontal Line Test. Itis one-to-one.g�x� � �x � 6

−9

−2

9

10

133.

f �1�x� � x2 � 10 , x ≥ 0

x2 � 10 � y

x2 � y � 10

x � �y � 10 , y ≥ �10, x ≥ 0

y � �x � 10, x ≥ �10, y ≥ 0

f �x� � �x � 10 134.

f�1�x� �96 � x2

16, x ≥ 0

y �96 � x2

16

16y � 96 � x2

x2 � 16�6 � y� � 96 � 16y

x � 4�6 � y, y ≤ 6, x ≥ 0

y � 4�6 � x

f �x� � 4�6 � x, x ≤ 6, y ≥ 0

129.

f�1�x� � 2x � 10

y � 2�x � 5�

x � 5 �12

y

x �12

y � 5

y �12

x � 5 130.

f�1�x� �17

�8x � 3�

8x � 3 � 7y

8x � 7y � 3

x �18

�7y � 3�

y �18

�7x � 3�

f �x� �7x � 3

8

131.

f �1�x� � 3�x � 34

x � 3

4� y3

x � 3 � 4y3

x � 4y3 � 3

y � 4x3 � 3

f �x� � 4x3 � 3 132.

f�1�x� � 3�x � 25

x � 2

5� y3

x � 2 � 5y3

x � 5y3 � 2

y � 5x3 � 2

135. Negative correlation 136. No correlation ©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 85: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Review Exercises for Chapter 1 85

137. (a)

(b) Yes, the relationship is approximately linear.Higher entrance exam scores, are associatedwith higher grade-point averages, y.

x,

Exam score

Gra

de-p

oint

ave

rage

y

x65 70 75 80 85 90 95

1

2

3

4

138. (a)

(b) Answers will vary.

Distance bent(in centimeters)

Tim

e to

fai

lure

(in

hour

s)

x

y

5 10 15 20 25 30 35

10

20

30

40

50

60

139. (a)

Time (in seconds)

Spee

d (i

n m

eter

s pe

r se

cond

)

t1 2 3 4

5

10

15

20

25

30

35

40

s (b) (Approximations will vary.)

(c)

(d) For m/sec.t � 2.5, S � 24.7

s � 9.7t � 0.4; 0.99933

s � 10t

140. (a)linear model;

(b)

0 483.6

4.3

�0.91997y � �0.0119t � 4.164,

141. y � 95.174x � 458.423 142.

100

04.5 6.5

y � 95.174x � 458.423

143. The model does not fit well. 144. No. The data stops at �6.00, 100.0�.

145. False. and g��1� � �52 � 28g�x� � ���x � 6�2 � 3� � ��x � 6�2 � 3

146. True. odd f �1�x� � x1�n, n 147. False. or satisfies f � f �1.f �x� � x f �x� �1x

148. False. The slope can be positive, negative, or 0.

(c)

(d), (e) Answers will vary.

0 483.6

4.3

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.

Page 86: PART I CHAPTER 1 Functions and Their Graphs...3. 2 2 4 4 6 6 8 8 10 10 (2, 3) x y m = −3 m = 1 m = 2 m = 0 4. –6 –2 –4 –2 2 4 (−4, 1) x m = m = −3 m = 3 1 2 m is undefined

Chapter 1 Practice Test

86 Chapter 1 Functions and Their Graphs

1. Find the slope of the line passing through the points and �1, 3�.��2, 2�

10. Determine the open interval(s) on which the function is increasing.f �x� � 12x � x3

2. Find an equation for the line passing through the points Use a graphing utility to sketch agraph of the line.

�3, �2� and �4, �5�.

3. Find an equation of the line that passes through the point and has slope Use a graphing utility tosketch a graph of the line.

�3.��1, 5�

4. Find the slope-intercept form of the line that passes through the point and is perpendicularto 3x � 5y � 7.

��3, 2�

5. Does the equation represent y as a function of x?x4 � y4 � 16

6. Evaluate the function at the points x � 0, x � 2, and x � 4.f�x� � �x � 2���x � 2�

9. Use a graphing utility to sketch the graph of the function and determine if the function is even, odd,or neither.

f�x� � 3 � x6

11. Use a graphing utility to approximate any relative minimum or maximum values of the function y � 4 � x � x3.

7. Find the domain of the function f�x� � 5��x2 � 16�.

8. Find the domain of the function g�t� � �4 � t.

12. Compare the graph of with the graph of y � x3.f�x� � x3 � 3

13. Compare the graph of with the graph of y � �x.f �x� � �x � 6

14. Find if and What is the domain of g � f ?g�x� � x2 � 2.f�x� � �xg � f

15. Find if and What is the domain of f�g?g�x� � 16 � x4.f �x� � 3x2f�g

16. Show that and are inverse functions algebraically and graphically.g�x� �x � 1

3f�x� � 3x � 1

17. Find the inverse of Graph f and in the same viewing rectangle.f�1f �x� � �9 � x2, 0 ≤ x ≤ 3.

18. Use a graphing utility to find the least squares regression line for the points Graphthe points and the line.

�4, 5�.�3, 3�,�0, 1�,��1, 0�,

©H

ough

ton

Miff

lin C

ompa

ny. A

ll rig

hts

rese

rved

.