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 =...
TRANSCRIPT
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©
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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
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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
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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�
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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�
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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
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Com
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. All
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s re
serv
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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
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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
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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�,
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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,
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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
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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
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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�
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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
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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
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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
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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
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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
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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,
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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
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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�
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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
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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
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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
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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�
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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�
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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
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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
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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
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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�
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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
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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�
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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
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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
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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�
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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
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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�
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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
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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
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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
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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�
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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
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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�
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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
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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
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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
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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
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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�
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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
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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��
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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�
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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 �
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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�
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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
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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
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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
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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
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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
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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.
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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
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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
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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,
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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.
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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
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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.
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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
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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
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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
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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
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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
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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.
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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�
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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
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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�,
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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.
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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�
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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�,
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