1.1 exercises - deer valley unified school district / …•/checkpoint now try exercise 57(b)....

30 C h a p t e r ! Functions and Their Graphs

Example? Equations of Perpendicular Lines

Find ihe slope-intercept form of the equation of the l ine thai passes through the

poin t (2. — 1 ) and is perpendicular to the l i n e

S o l u t i o n

From Example (i. you know that the equation can be wr i t t en in the slope-intercept

form v = ; x - i You can see that the line has a slope of 5. So. any l ine

perpendicular to th i s line must have a slope of -5 (because -5 is the negative

reciprocal of \). So, the l ine through the point (2. - 1 ) has the following equation.

- 3 ' Si:i i] i lu>

V - - .V + 2 \Vrilc ::i < :P; -C- : : - ;J :XO;- ; :.-ri::

The graphs of both equations are shown in Figure 1 . 1 1 .

'•/CHECKPOINT Now try Exercise 57(b).

Figure 1.11

Examples Graphs of Perpendicular Lines

Use a graphing ut i l i ty to graph the lines

S o l u t i o n

If the v iewing window is nonsquarc. as in Figure 1 . 1 2 . the two lines w i l l not

appear perpendicular. If. however, the viewing window is square, as in Figure

i . I 3. the lines wi l l appear perpendicular.

Figure 1.12

:/CKfCf.'po/wr Now trv Exercise 67.

Figure 1.13

Section 1 .1 Lines in the Plane

1.1 Exercises

Vocabulary Check

£ Match each equation w i t h i t s form

(a) Av + B.v - (' - U

(b) -v = " -

(c) i = ''

i d ) i - "» " "

(e) V - V; = "HA - T| I

in Exercises 2-5. fill in the blanks.

2. For a line, the ral iu of the change in y tn the change in i is called the _

3. Two lines are __ i t and only if [heir slopes are equal .

4. Two lines are _ if and only i f their slopes arc n e g a t i v e reciprocals of t

5. The prediction method ___ is Ihe method used to e s t ima te a p o i n t on a Ilie between the given points

In Exercises 1 and 2. identify the line ihul has theindicated slope.

1. (a) in - t t h ) in is undefined (c) »< -- - 2

2. {&) in = 0 ( h i m - - ^ i c} !u - i

In Exercises 7-10. find the slope ol Ihe line passing throughthe pair of points. Then use a graphing utiiit} to pint thepoints and use the draw feature to graph ihe line segmentconnecting the two points. (Use a square ti'iniig.)

7. (0. - l ( ) i . ( 4. Oi 8. !l 4 i . •-. - 41

in Kxercises 1 1-18. use the point on the line and the slope ofthe line to lind three additional points through which thelinn passes. (There are many correct answers.*

1'ninl Slop,'

In Exercises 3 and 4. sketch the lines through the point w i t hthe indicated slopes on the same set of coordinate axes.

Point Sln/m

3. 12.3) (a) (I (b) 1 ( e i 2 i d ' 3

•*• I--*. I ) (a ) I I b ) -5 l c l ; , d j fndct lned

In Exercises 5 and 6, estimate the slope of the line.

-> 6. -

16.

17.

IS.

In Exercises 19-24. t a l (ind the slope and v-intcrcupt (ifpossible) of the equation of the line algebraic-ail), and ( b /sketch the l ine by hand. Use a graphing ut i l i t ) to verify youranswers to parts (a) and I b l .

12 Chapter 1 Functions and Their Graphs Section I . I Lines in the Plane 13

In Exercises 25-32, find the general form of the equationof the line that passes through the given point and has theindicated slope. Sketch the line hy hand. Use a graphingutility to verify your sketch, if possible.

I'ninl Slii/v

25. {(I. -2l in -- i

26. ( - 3 . 6 )

27. (2. -3)

28. (-2, -51

29. (ft, - 1 1

30. (- 10.41

31. (-i. ;]

32. (2.3. -S.5)

In Exercises 33-42, find the slope-intercept form of theequation of the line that passes through the points. I'sc ;igraphing utility to graph the line.

33. (5. - I). (-5. 51

34. 14. .1). (-4. -41

35. ( -« . 1 ) , (-8. 71

36. 1 -1 .4 ) . (ft . 4)

*7. U.i).(lJ)3 8 . ( I , I ) , ( f t . --{I

39. i-i-s).(i-; ji40. (j. 3). ( - j . i l

41. (1.0.6). 1-2, -0.6).

42. (-8. 0.61. (2. -2.4)

In Exercises 43 and 44. find the slope-intercept fornj of theequation of the line shown.

43.

45. Annual Salary A jeweler's salary \vtis S2S.500 in 2004and S32.WO in 2006. The jeweler's salary follows a l i n e a rgrowth pattern. What wi l l the jeweler's salary be in 200N1'

46. Annual Salary A librarian's salary1 was S25.000 in 2004and S27.500 in 2006. The l ibrar ian 's salary follows a l ineal-growth pattern. What wi l l the librarian's salary he in 2008'

In Exercises 47-50, determine the slope and j-intercept of thelinear equation. Then describe its graph.

47. ,< - 2v = 4

48. 3.< -I 4y = 1

49. v = -ft

50. > = 12

In Exercises 51 and 52, use a graphing utility to graph theequation using each of the suggested viewing windows.Describe the difference between the two graphs.

51. v = 0.5.V - 3

I'ninl

5 7 . ( 2 . I )

58. (-3. 2)'

59. (-!, I)

60. ( -3 .9 , - 1 .41

61. (3. -2)

62. (-4. 1 )

Xmin = -5 Xmin - -2Xmax = 10 '• Xmax = 10

I Xsel = I , Xsel = IYmin - - 1 : Ymin = -4Y'max - 10 ! Ymax = 1Ysel = 1 : I Ysel = 1

52. v = - S.v + 5

Xmin = -5 , Xmin = -5Xmax ~ 5 ! Xmax = 10

! Xsel = 1 Xsel = 1Ymin = -10 ; Ymin = -80

1 Ymax =10 ' Ymax =80Ysel = I Ysel = 20

f

^

it1

"I*

",

In Exercises 53-56, determine whether the lines £., and JU 1passing through the pairs of points are parallel, perpendi-cular, or neither.

In Exercises 57-62, write the slope-intercept forms of theequations of the lines through the given point (a) parallel tothe given line and (b ) perpendicular to the given line.

In Exercises 63 and 64, the lines are parallel. Kind the .slope-intercept fVirni of the equation of line v , .

63. ' 64.

In Exercises 65 and 66, the lines are perpendicular. Find the.slope-intercept form of the equation of line.v-..

65. ' 66. .'

Graphical Analysis In Exercises 67-70, identify ;m>relationships that exist among the lines, and then use ;;graphing utility to graph the three equations in the sameviewing window. Adjust the viewing window so that eachslope appears visually correct. Use the slopes of the lines toverify your results. i e s In ie rpre i the meaning o i ' the slope of ihe cqi;::;; •

part i h ) in ihc cnmexi of ihc prohlcni.

d) I sL the equa t inn from part ( h ) in c M i m a t L i lf .ood\ ii Fin. in UIL \ i i 201(1 Oo im i h n ti L L i i r i tc L s l i n i i l ion I \p l t i n

73. !JLU>ht Tht - MSL ui i i i n i i t io «\L mntL i L t L i m i n L . th<- S t C U p l l L S S l l f t l lL T O O i TllL 11SL Ul L

hi ion! in the I Iuure is 3 to 4. D e t e r m i n e hI I L I - ^ i i i i l l . til t il h il IK h L

74. Road Grade When t i m i n g douii a n i o u n t a i r i :'»:>d.notice waming signs indieatiny thai it is a "12 ' - ^raThis means lh; t t I!K* slope of tlic road is -j^, Approx i' l ie amount ol hnri/onlal chanii: in vour posi i in: ; i inote from elevation markers t h a i ; - u '- \- dfscei^eJfeet \ e - r t i c a l l v .

Rate of Change In Exercises 75-78. you are yiven tliedollar vahie of a product in 2006 and the rate at which thevalue of the product is expected to change during the next5 years. Write a linear equation that gives the dollar valuei of the product in terms of thi? \i*ar t. ( L e t t = (i represent

Huts

$4.50 increase per > c a r

S2000 decrease per veai

$5600 decrease per > ear

Graphical Interpretation In Exercises 79-82, match !hedescription with its graph. Determine the slope of eachgraph and how i; is interpreted in the given context. [Thegraphs, are labeled la], i n ) , ( c i , and id) . ]

83. Depreciation \l district purchases'a h i g h - v o l u m e

pnntLT, copier, and scanner tor $25,000. Af te r !0 vear.v theeu'-sipment w i l l have to be replaced, i ts v a l u e at t h a i t ime ssexpected to he $2000.

• a W ri te a l inea r equa t ion g iv ing [he v a l u e I ' o! thee q u i p m e n t d u r i n g the 10 years i ( w i l l be used

i b : Use a graphing u i i l i u to g i a p h the l i n c a i equat ionrepresenting the depreciation of the e q u i p m e n t , and usethe vuhit' or tmce feature to complete the [able.

F i (V

•S5. Cost, Revenue, and Profit A contractor purchases ahulldo/er for $36,500 The bulkio/.er requires an aver geexpend i tu re of $5.25 per hour for fuel and ma in t enanceand the operator is paid S I 1.50 per hour

. a ' Wr i t e a l inear equa t ion g i v i n g the tola! cost (_' ..if oper-a t ing the bulklo/er for / hours. ( I n c l u d e the nurch isecost of the bulldo/cr )

iln Assuming that customers are charged $27 per hour ofbul ldo/er use. wr i te an equa t ion Tor the r e \ enuc R

derived from / hours of use

id L'se the profi t f o rmula !/J ~ K - C\o w r i t e an

equat ion for the profit de r ived Irom / h o u i s n!' use

( d i Use the resul t of pan i c ) to find the break-even point( t h e number of hours the bulldo/er must he used tovield a profit of 0 dollars)

S6. Rental Demand A real estate office handles an aparunentcomplex wi th 50 uni ts . When the rent per u n i t is S3 SO permonth, a l l 50 u n i t s are occupied. However, when the rent isSfi25 per month, the average number of occupied unit-.drops to 47. Assume that the relationship between themonthk rent /? and the demand .v is l inear .

( a ) Write the equation of the l ine g iv ing the demand .v in

terms ol the rent p.

i b i L'se a graphing u t i l i t y to graph the demand equat ionand use the trace feature to estimate the numbci oru n i t s occupied when the rent is $655. Verify youranswer a lgebra ica l ly .

i c ) Use the demand equation to predict the numbe i ot unitsoccupied when ite'tent is lowered to $595. Verify youranswer graphically.

87. Education In 1991 . Perm State U n i v e r s i t y had inen ro l lmen t oi"75.34*3 students . By 2005. the e n r o l l m e n t nadincreased to" 80.124. .^ . . . - ' ; . v '- . ; : , "

True t>r ftitse'.' In Exercises 89 and 90, determine whetherthe statement is Jrue or false. Justify your answer.

89. The l ine through 1 - S. 2) and i - 1 . 4 ) and the l ine throughi ( ) . - 4 i and ( - 7. 7t are p a r a l l e l .

90. If the points I 10. -31 and (2. - <-)} l i e on the same l i n e , t h e n

the point 1 - 1 2 , - - V - J also l i e s on that l i n e

Exploration In Exercises 91-94. use a graphing u t i l i t y to[jraph (he equation of the line in the form

Lse the graphs to make a conjecture about what a and brepresent. Verify your conjecture.

94. -; - 7 - 1

In Exercises 95-98. use the results of Exercises 91-94 towri te an equation of the line that passes through the points.

95. A-miercepi : [ 2 . 0 ) 96. .v-intcrcept: I - 5 . 0 '

. - i n t e r c e p t : ( 0 . 3 ) v-irnerccpt. (0. 4}

f>7. i -m ie r cep t : 1 - ^ . O j 98. .r- intcrcepf {-,.()}

'• .ntei 'cepi: 10, -?} • • . - in te rcep t : ( ( ) . ; !

; \'.-.\\\\\K In Exercises 99 and 100,detenniiiL' which equation(s) may he represented by thegi'tiph show n, (There may he more than one correct answer.)

<w- ' 100.

Section 1.1 Lines in the Plane 15

i i h r ; ; r v u! { 'a rc ' :* ]• i i ; i c < i o j = v. In Exercises 101 and 102.determine which pair of equations may he represented b>the graphs shown.

101. 102.

103. Think About It Doe1, even, line haveand ay-intercept' . ' E x p l a i n .

10-4. Think About It Can ever, l i nslope-intercept f o r m ? E x p l a i n .

1(15. Think About It Does every l ine have an i n f i n i t e n u m b e iof lines that are parallel to the given line1.1 Explain.

106. Think About it Does every l ine have an in I] ru le numberof l ines t ha t are perpendicular 10 the g iven l i n e . ' E x p l a i n .

Skills Review

In Exercises 107-112, determine whether the expression is apolynomial. If i( is, write the polynomial in standard form.

108. AV Kh : • 1

110. 2v : 2r v ' - "*

In Exercises 1 1 3 - 1 1 6 , factor the trinomial.

1 1 3 . ,v: - 6v - 27 114. .r -- i 1

115 . 2v - -f- | U - 40 l l f i . I (\\. Make a Decision To work an extended :ipplic;ition

analyzing the numbers of bachelor's degrees enrned bywomen in the United States from 1985 to 2005 visi t rhistextbook's Online Study Center. : • - . .• • '•

24 Chapter 1 Functions and Their Graphs

12 Exercises

Vocabulary Check

Fill in the blanks.

1. A relation that assigns to each element x from a set of inputs, or.in a set of outputs, or , is called a .

2. For an equation that represents y as a function of x, the _and the variable is the set of all y in the range.

[*2- 4,x < 0 .

., exactly one element y

. variable is the set of all x in the domain,

3. The function f(x) — \~ ^ * « is an example of a function.

4. If the domain of the function/is not given, then the set of values of the independent variable for whichthe expression is defined is called the

u

5. In calculus, one of the basic definit

In Exercises 1-4, does the relatioiExplain your reasoning.

1. Domain Range 2. Dom

2 C r\ ~" L,

2 2

3. Domain Range 4. Dom

National /-Cubs (Ye'T •\—>- PiratesLeague \

xDodgers

mAmerican /<0rioles 19'League ^Yankees 20C

^ Twins 20(20(20(20(20(

In Exercises 5-8, decide whether tas a function of*. Explain your reas

Input,* -3 -i 0

Output, y -9 -1 0

h

i aescnoe a runcnon: o. ^^ Q j 2

am /?an#e OutPut'^ -4 -2 0 :

^--' '4 Input,* 10 7 4 7_ »?• I '

^^-j;5 Output, y 3 6 9 12

^ 8' Input,* 0 3 9 12c«« Range, . . . . , . . Output, y 3 3 3 3

ir) (Number of F

11. is tiyea

0 the

I 4 12. Letyea

10 In Ex«

15

1<V V =

15 17. 2x

3 19. y2

North Atlantic "1" -v "

tropical storms in Exercises 9 and 10, which sets of ordered pairs representand hurricanes) functions from A to B? Explain.

|9^XCZ14 9. A = (0, 1,2, 3} and 5 = {-2, -1,0, 1,2} functic

)0 /, 15 ra^ {rn_ n ,n , -9M2,o), (3. 2)} SimPli

)1 -7^ 16 (b) {(0, - D, (2, 2), (1, -2), (3, 0

)2 /?/ 26 (c) ((0' 0), (1, 0), (2, 0), (3, 0)}

)3// (d) {(0,2), (3,0), (1,1)}\A / >/^ X W. A = {a, b, c} and B = {0, 1, 2, 3}

(a) {(a, 1), (c, 2), (c, 3), (ft, 3)}

le relation represents y (b) {(a, 1), (b, 2), (c, 3)}

ionin§- (c) {(1, a), (0, a), (2, c), (3, b)}

3 (d) {(c, 0), (b,Q), (a, 3)}

9

, (1, D} ,5 h

(aj

(c '

26. g(

(a

(b

(c

( d

Section 1.2 Functions 25

>resent

Circulation of Newspapers In Exercises 11 and 12, use thegraph, which shows the circulation (in millions) of daily

in the United States, i Source: Editor &

co

'£c

cd"3oU

60-

50-

40-

30-

20-

r==38f_ _ _ '_ • Morning

9 Evening

B"• »---*- •* * v —g — 4 s -

A. 1 1 1 L 1 1 1 ..1 L- >

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

11. Is the circulation of morning newspapers a function of theyear? Is the circulation of evening newspapers a function ofthe year? Explain.

12. Let f(x) represent the circulation of evening newspapers inyear A;. Find/(2004).

In Exercises 13-24, determine whether the equationrepresents y as a function of x.

13. A-2 + y2 = 4

17. 2x + 3y = 4

19. y2 = A-2 - 1

21. y = |4 - x\. x= -7

14. x = y2 + 1

16. y = -Jx + 5

18. x = -y + 5

20. A- + y2 = 3

22. |y| = 4 - x

24. y = 8

In Exercises 25 and 26, fill in the blanks using the specifiedfunction and the given values of the independent variable.Simplify the result.

(a) /(4) =

(c) /(4r) = (d) f(x + c) =) + l

26.

-2

) 2 -2 () 2 - 2 f

In Exercises 27-42, evaluate the function at each specifiedvalue of the independent variable and simplify.

27. /(/) = 3f + 1

(a) /(2)

28. g(y) = 7 - 3y

(a) g(0)

29. h(f) = t2-2t

(a) h(2)

30. V(r) = firr3

(b) /(-4)

(b) g(i)

(b) /i(1.5)

(c) f(t + 2)

(c) g(s + 2)

(c) h(x + 2)

(b) V(f) (c) V(2r)

31. /(y) = 3 - vy

(a) /(4) (b) /(0.25) (c) /(4;c2)

32. f(x) = VATT8 + 2

33. q(x) =2 -

(a) q(0)~ -, ,

(a) ?(2)

(b) /(I)

(b)

(b)

= \A35. f(x) =

(a) /(3)

36. f(x) = \ /(4)

37.

(a) /(-1)

38. f(x) =

39. /(x) =

40. f(x) =

41./U)=

+ 1, A: < 0+ 2, A; > 0

(b) /(O)

+ 5, A- < 02 - ;c2, A: > 0

i (b) /(O)x2 + 2, A- < 12*2 + 2, A; > 1

(b) /(I)

A-2 - 4, A: < 01 - 2A:2, A: > 0

2) (b) /(O)

(A: + 2, x < 014, 0 < x < 2[x2 + 1, A- > 2

2) (b) /(I)

(c) /(A: - 8)

(c) q(y + 3)

(c) q(~x)

(c) /fo

(c) f(t)

(c) /(2)

(c) /(I)

(c) /(2)

(c) /(I)

(c)


f5 - 2*. * < 042./to = 5 , 0 < * < 1

14* + 1, x>l

(a) /(-2) (b) /(|) (c) /(I)

In Exercises 43-46, complete the table.43. h(t) = \ + 3|

t

h(t)

-5 -4 -3 ^ -2 -1

44. /(,) =- 2

s

/(*)0 1 1

52 4

45. 4>

46.* - 3, > 3

x

h(x)

1 2 3 4 5

In Exercises 47-50, find all real values of x such that/to = 0.

47. /(*) = 15-3*

49. /(,) = -i

48. /W = 5* + 1

50. U) = .Zi

In Exercises 51 and 52, find the value(s) of x for which/to = *to-

51. /to = x-, g(x) = x + 2

52. /to = ,t2 + 2* + 1, g(x) = 7x - 5

In Exercises 53-62, find the domain of the function.

53. f(x) = 5x2 + 2x - 1 54. g(x) = 1 - 2x2

55. h(t) = - 56. ^(v) =3

62./(*) =

In Exercises 63-66, use a graphing utility to graph thefunction. Find the domain and range of the function.

63. f(x) =65. g(x) =

64. f(x) = V^T

66. g(x) = \x-5\n Exercises 67-70, assume that the domain of / is

the set A = {-2, -1,0,1,2}. Determine the set of orderedpairs representing the function/.

67. f(x) = x- 68. f(x) =x2-3

69. f(x) =\x\2 70. f(x) = \ + 1|

71. Geometry Write the area A of a circle as a function of itscircumference C.

72. Geometry Write the area A of an equilateral triangle as afunction of the length s of its sides.

73. Exploration The cost per unit to produce a radio model is$60. The manufacturer charges $90 per unit for orders of100 or less. To encourage large orders, the manufacturerreduces the charge by $0.15 per radio for each unit orderedin excess of 100 (for example, there would be a charge of$87 per radio for an order size of 120).

(a) The table shows the profit P (in dollars) for variousnumbers of units ordered, x. Use the table to estimatethe maximum profit.

•f Units,*

110

120

130

140

150

160

170

Profit, P

3135

3240

33153360

3375

- 3360

3315

(b) Plot the points (x, P) from the table in part (a). Does therelation defined by the ordered pairs represent P as afunction of xl

(c) If P is a function of *, write the function and determineits domain.


raph the

of / isF ordered

tion of its

ingle as a

) model isorders ofmfacturerit orderedcharge of

>r variousi estimate

Does thent P as a

letermine

74. Exploration An open box of maximum volume is to bemade from a square piece of material, 24 centimeters on aside, by cutting equal squares from the corners and turningup the sides (see figure).

(a) The table shows the volume V (in cubic centimeters) ofthe box for various heights x (in centimeters). Use thetable to estimate the maximum volume.

f Height, x

123456

Volume, V

4848009721024

980864

(b) Plot the points (x, V) from the table in part (a). Does therelation defined by the ordered pairs represent V as afunction of xl

(c) If V is a function of x, write the function and determineits domain.

(d) Use a graphing utility to plot the point from the table inpart (a) with the function from part (c). How closelydoes the function represent the data? Explain.

x « 24 -2x « x

75. Geometry A right triangle is formed in the first quadrantby the x- and y-axes and a line through the point (2, 1) (seefigure). Write the area A of the triangle as a function of x,and determine the domain of the function.

76. Geometry A rectangle is bounded by the .x-axis and thesemicircle y = V36 - x2 (see figure). Write the area A ofthe rectangle as a function of x, and determine the domainof the function.

77. Postal Regulations A rectangular package to be sent bythe U.S. Postal Service can have a maximum combinedlength and girth (perimeter of a cross section) of 108 inches(see figure).

(a) Write the volume V of the package as a function of x.What is the domain of the function?

(b) Use a graphing utility to graph the function. Be sure touse an appropriate viewing window.

(c) What dimensions will maximize the volume of thepackage? Explain.

78. Cost, Revenue, and Profit A company produces a toy forwhich the variable cost is $12.30 per unit and the fixedcosts are $98,000. The toy sells for $17.98. Let x be thenumber of units produced and sold.

(a) The total cost for a business is the sum of the variablecost and the fixed costs. Write the total cost C as afunction of the number of units produced.

(b) Write the revenue R as a function of the number ofunits sold.

(c) Write the profit P as a function of the number of unitssold. (Note: P = R- C.)


Revenue In Exercises 79-82, use the table, which showsthe monthJy revenue y (in thousands of dollars) of alandscaping business for each month of 2006, with x = 1representing January.

Month, x

123456789101112

Revenue, .y

5.25.66.68.311.515.812.810.18.66.94.52.7

A mathematical model that represents the data is

f( ) = {-1-97*+ 26.3JW (0.505x2 - 1.47x + 6.3'

79. What is the domain of each part of the piecewise-definedfunction? Explain your reasoning.

80. Use the mathematical model to find /(5). Interpret yourresult in the context of the problem.

81. Use the mathematical model to find/(11). Interpret yourresult in the context of the problem.

82. How do the values obtained from the model in Exercises 80and 81 compare with the actual data values?

83. Motor Vehicles The numbers n (in billions) of milestraveled by vans, pickup trucks, and sport utility vehicles inthe United States from 1990 to 2003 can be approximatedby the model

() = [~6-I3f2 + 75-&t + 577' 0 < ; < 6~ J24.9r + 672, 6 < t < 13

where t represents the year, with t = 0 corresponding to1990. Use the table feature of a graphing utility to approx-imate the number of miles traveled by vans, pickup trucks,and sport utility vehicles for each year from 1990 to 2003.

IfWl~ 1000-g 900-

3 800-

•£ 700-

•3 600-

1$ 500-

> 400-

£ 300 HJ> 200-

§ 100-

- gtajifa— - • ;•-I

— — r~-j ——

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

Year (Oo 1990)Figure for 83

84. Transportation For groups of 80 or more people, acharter bus company determines the rate per personaccording to the formula

Rate = 8 - 0.05(n - 80), n > 80

where the rate is given in dollars and n is the number ofpeople.

(a) Write the revenue R of the bus company as a functionof n.

(b) Use the function from part (a) to complete the table.What can you conclude?

n

R(n)

90 100 110 120 130 140 150

(c) Use a graphing utility to graph R and determine thenumber of people that will produce a maximumrevenue. Compare the result with your conclusion frompart (b).

85. Physics The force F (in tons) of water against the face ofa dam is estimated by the function

F(y) = 149.76VlOv5/2

where y is the depth of the water (in feet).

(a) Complete the table. What can you conclude from it?

y

F(y)

5 10 20 30 40

(b) Use a graphing utility to graph the function. Describeyour viewing window.

(c) Use the table to approximate the depth at which theforce against the dam is 1,000,000 tons. How could youfind a better estimate?

(d) Verify your answer in part (c) graphically.


6. Data Analysis The graph shows the retail sales (inbillions of dollars) of prescr

T States from 1995 through 2004sales in year x. < Source: NaDrugstores. '

fix)

^ 240 -- ^-

.2 200 -- -C_>^-M O

7-Wf GG O

•3 rs co 1 9Q

u 1 • *OS a 80-- -§ •

.5, 40 -|-eople, a 1 A

person 1996 1998

(a) Find/(2000).

mberof /(2004) -/(1995)

iption drugs in the United •"•• j v > t< t _ j •. Let/(jr) represent the retail . ,, , ., ,

Synthesis •

~ Jrwe or Fafae ? In Exercises 93 and 94, detei* the statement is true or false. Justify your an

% * 93. The domain of the function f(x) = x4 —...» and the range of f(x) is (0, 00).

94. The set of ordered pairs {(-8, -2), (-(-2, 2), (0, 4), (2, - 2)} represents a funct

1 x ^U-Tarv oi Par2000 2002 2004 . : , -piecewise-defin

Year95. )

5-4i

(b)rmd 2004-1995 X;

function ancj jnterpret tne result in the context of the problem. / l -

(c) An approximate model for the function is \_3_7-i .16 table. ,J4 0) _, .

PW = -0.0982;3 + 3.365^ - 18.85r + 94.8, *- _ 3 _s < t < u

where P is the retail sales97. -(in billions of dollars) and t

represents the year, with t = 5 corresponding to 1995. 10 -

line the • data in the ^aPh- \ 6 -Aimum>nfrom < 5 6 7 8 9

P(r)

0 11 12 13 14 (-2,4) ^-

-6 -4 -2

;ni runcaons in Jixercises •ed function for the graph sh(y

96. .1

. (0, 4) 3 -

- \1'0)/4,o) /

1 i 1 > r 9 /* 1 1 »• •* " /- 1 ~

- \"- \ -3-

98.

F \::" (5'6),/ Ly ("2'4)\) , , \ ' /

-6-4 -2/

2 4 6 (O.W

•mine whetherswer.

1 is ( — 00, oo),

6,0), (-4,0),ion.

)5-98, write a»vn.

^' (2, 3) y,

/70,1)

. 1 2 3

vk

;

tT^1' , r. 2 4>T

(6,-l)

-nit?

(d) Use a graphing utility to graph the model and the datain the same viewing window. Comment on the validityof the model.

99. Writing In your own words, explain the meanings ofdomain and ranee.

100. r/zj'nJk About It Describe an advantage of functionJ" In Exercises 87-92, find the difference quotient and simplify notation.

your answer.Skills Review

In Exercises 101-104, perform the operation and simplify.

4 3 . x1

;scribe *

ch the JId you

-3&4l

87 f(t) - n J(X + C) ~f(x) ^

C

h

89 fir) -> /(2 + /i) — /(2)' A

90. f(r\ T 3 + .. /U + /l) ~/W ,

101. 12 - 102.

103.

104.

2x3 + 1 lx2 -5x

2 + A: - 20

+ 10

- 5

2x- + 5x - 3-., \ -., "7A ~T / A /

2U - 9) " 2(x - 9)

symbol J indicates an example or exercise that highlights algebraic techniques specifically used in calculus.


1.3 Graphs of Functions

The Graph of a FunctionIn Section 1.2, functions were represented graphically by points on a graph in acoordinate plane in which the input values are represented by the horizontal axisand the output values are represented by the vertical axis. The graph of a func-tion/is the collection of ordered pairs (x,f(x)) such that x is in the domain of/.As you study this section, remember the geometric interpretations of A: and/Oc).

x = the directed distance from the ;y-axis

/(jc) = the directed distance from the ;t-axis

Example 1 shows how to use the graph of a function to find the domain andrange of the function.

Example 1 Finding the Domain and Range of a Function

Use the graph of the function / shown in Figure 1.19 to find (a) the domain off,(b) the function values/(— 1) and/(2), and (c) the range of/.

Range

(-1.-5)Domain

Figure 1.19

Solution

a. The closed dot at (— 1, — 5) indicates that x = — 1 is in the domain of /whereas the open dot at (4, 0) indicates that x = 4 is not in the domain. So, thedomain of/is all x in the interval [ — 1 , 4 ) .

b. Because (— 1, — 5) is a point on the graph off, it follows that

/(-I) = -5.

Similarly, because (2, 4) is a point on the graph off, it follows that

/(2) = 4.

c. Because the graph does not extend below/(— 1) = — 5 or above/(2) = 4, therange of/is the interval [—5, 4].

CHECKPOINT Now try Exercise 3.

What you should learnFind the domains and ranges of functionsand use the Vertical Line Test for functions.Determine intervals on which functionsare increasing, decreasing, or constant

minimumvahies of functions.Identify and graph step functions and

shaatd team itGraphs of functions provide a visual relation-

Exercise 88 on page 40, you will use tie graphof a step function to modelthe cost ofsend-

Ja^cfeige,

Stephen Chemin/Getty Images

STUDY TIP

The use of dots (open or closed)at the extreme left and rightpoints of a graph indicates thatthe graph does not extendbeyond these points. If no suchdots are shown, assume that thegraph extends beyond thesepoints.

Section 1.3 Graphs of Functions 31

fcample2 Finding the Domain and Range of a Function

find the domain and range of

f(x) = v^4.

Algebraic SolutionBecause the expression under a radical cannot benegative, the domain of f(x) = ^/x - 4 is the set

all real numbers such that x — 4 > 0. Solves linear inequality for x as follows. (For help

"lirith solving linear inequalities, see Appendix E.)

j t - 4 > 0

x> 4

Write original inequality.

Add 4 to each side.

the domain is the set of all real numbersater than or equal to 4. Because the value of a

expression is never negative, the range ofjr) = ~Jx - 4 is the set of all nonnegative real

Bbers.

Now try Exercise 7.

Graphical SolutionUse a graphing utility to graph the equation y = V* ~~ 4, as shownin Figure 1.20. Use the trace feature to determine that thejr-coordinates of points on the graph extend from 4 to the right.When x is greater than or equal to 4, the expression under theradical is nonnegative. So, you can conclude that the domain is theset of all real numbers greater than or equal to 4. From the graph,you can see that the ^-coordinates of points on the graph extendfrom 0 upwards. So you can estimate the range to be the set of allnonnegative real numbers.

Figi

y = t/x-4

>--1

jre 1 .20

By the definition of a function, at most one _y-value corresponds to a given~e. It follows, then, that a vertical line can intersect the graph of a function

t once. This leads to the Vertical Line Test for functions.

al Line Test for Functions

? points in a coordinate plane is the graph of y as a function of x ifif no vertical line intersects the graph at more than one point.

pie 3 Vertical Line Test for Functions

(the Vertical Line Test to decide whether the graphs in Figure 1.21 represent^ a function of x.

Ion

&is is not a graph of y as a function of x because you can find a vertical lineintersects the graph twice.

• is a graph of y as a function of jr because every vertical line intersects the" i at most once.

Now try Exercise 17.

(a)

(b)

Figure 1.21


Increasing and Decreasing FunctionsThe more you know about the graph of a function, the more you know about thefunction itself. Consider the graph shown in Figure 1.22. Moving from left toright, this graph falls from x = — 2 to x = 0, is constant from x = 0 to x = 2, andrises from x = 2 to x = 4.

Increasing, Decreasing, and Constant Functions

A function/is increasing on an interval if, for any xl and x2 in the interval,

x} < x2 implies/(*,) < f(x2).

A function/is decreasing on an interval if, for any *, and x2 in the interval,

*, < x2 implies/U,) > f(x2).

A function/is constant on an interval if, for any *, and x2 in the interval,

/(*,) =/Ot2).

Example 4 Increasing and Decreasing Functions

In Figure 1.23, determine the open intervals on which each function is increasing,decreasing, or constant.

Solutiona. Although it might appear that there is an interval in which this function is

constant, you can see that if jc, < x2, then (.x,)3 < (x2)3, which implies that/(*,) < f(x2). So, the function is increasing over the entire real line.

b. This function is increasing on the interval (—00, — 1 ) , decreasing on theinterval (— 1, 1), and increasing on the interval (1, oo).

c. This function is increasing on the interval (-00, 0), constant on the interval(0, 2), and decreasing on the interval (2, oo).

f(x)=x3

-3

-2

la) •Figure 1.23

3 -4

(b)

,-,,,/ I

f(x)=x3-lx

Jr.Ml, -2)

4

TIP

Most graphing utilities aredesigned to graph functions ofx more easily than other typesof equations. For instance, thegraph shown in Figure 1.23(a)represents the equationx ~ (y ~ I)2 = 0. To use agraphing utility to duplicatethis graph you must first solvethe equation for y to obtainy = 1 ± -N/JC, and then graph thetwo equations y, = 1 + *J~x. andy2 = 1 — V* in the sameviewing window.

4 - -

3 - -

-2

Constant

Figure 1.22

/(*) = •ft:1-. -x + 3

0<x<2x>2

-2

" 2 I

(0, 1)

-2

(c)

/CHECKPOINT Now try Exercise 21.

pomhavior

of"•values

itiil

tfunciervaJ

The grap,'t'','.- a graphii* ' the pomi

J (0.6

Later, in• : >• the relati

Fie

I will fint Examp


Relative Minimum and Maximum ValuesThe points at which a function changes its increasing, decreasing, or constantbehavior are helpful in determining the relative maximum or relative minimumvalues of the function.

theand

(Definitions of Relative Minimum and Relative Maximum

A function value/(a) is called a relative minimum of/if there exists aninterval (x}, x2) that contains a such that

x, < x < x2 implies f(a) < f(x).

function value/(a) is called a relative maximum of/if there exists anral (*!, x2) that contains a such that

< x < x-, implies /(a) >

Figure 1.24 shows several different examples of relative minima and relativemaxima. In Section 2.1, you will study a technique for finding the exact points atwhich a second-degree polynomial function has a relative minimum or relativemaximum. For the time being, however, you can use a graphing utility to findreasonable approximations of these points.

Figure 1.24

Approximating a Relative Minimum

Use a graphing utility to approximate the relative minimum of the function given" = 3x2 - 4x - 2.

on

Tk^ graph of /is shown in Figure 1.25. By using the zoom and trace features of(ling utility, you can estimate that the function has a relative minimum atDt

i.67, -3.33). See Figure 1.26.

.in Section 2. 1 , you will be able to determine that the exact point at which[rive minimum occurs is (5, — $).

-3.28

K=.fififi91HB9 V= -3.3333320.62-3.39

Figure 1.26

0.71


Some graphing utilities have built-in programs thatamum or maximum values. These features are demonstrated in

When you use a graphing utilityto estimate the x- and y-values ofa relative minimum or relativemaximum, the zoom feature willoften produce graphs that arenearly flat, as shown in Figure1.26. To overcome this problem,you can manually change thevertical setting of the viewingwindow. The graph will stretchvertically if the values of Yminand Ymax are closer together.


Example 6 Approximating Relative Minima and Maxima

Use a graphing utility to approximate the relative minimum and relativemaximum of the function given byf(x) = -x3 + x.

Solution

The graph of/is shown in Figure 1.27. By using the zoom and trace features orthe minimum and maximum features of the graphing utility, you can estimate thatthe function has a relative minimum at the point

(-0.58, -0.38) See Figure 1.28.

and a relative maximum at the point

(0.58,0.38). See Figure 1.29.

If you take a course in calculus, you will learn a technique for finding the exactpoints at which this function has a relative minimum and a relative maximum.

^CHECKPOINT Now try Exercise 33.

Example 7 Temperature

During a 24-hour period, the temperature y (in degrees Fahrenheit) of a certaincity can be approximated by the model

y = 0.026*3 - \.03x2 + W.2x + 34, 0 < x < 24

where x represents the time of day, with x = 0 corresponding to 6 A.M.Approximate the maximum and minimum temperatures during this 24-hour period.

SolutionTo solve this problem, graph the function as shown in Figure 1.30. Using thezoom and trace features or the maximum feature of a graphing utility, you candetermine that the maximum temperature during the 24-hour period was approx-imately 64°F. This temperature occurred at about 12:36 P.M. (x ~ 6.6), as shownin Figure 1.31. Using the zoom and trace features or the minimum feature, youcan determine that the minimum temperature during the 24-hour period wasapproximately 34°F, which occurred at about 1:48 A.M. (x =» 19.8), as shown inFigure 1.32.

y = 0.026.x3 - 1.03.r2 + 10.2x + 3470 ' I/ 70 70

24luximum

nlK=e.6 • • - 24 0

Figure 1.30

•/CHECKPOINT

Figure 1.31


Figure 1.32

Figure 1.27

f(x) = -x3+x

3

2

AV"

Miniptum LL-2

Figure 1.28

-3

Maximum

-2

Figure 1.29

For instructions on how to use theminimum and maximum features,see Appendix A; for specifickeystrokes, go to this textbook'sOnline Study Center.

24


Iphing Step Functions and Piecewise-DefinedActions

of Parent Functions: Greatest Integer Function

jjjjireatest integer function, denoted by |xj and defined as the greatest*% less than or equal to x, has an infinite number of breaks or steps—

||ach integer value in its domain. The basic characteristics of the"^integer function are summarized below. A review of the greatestpfunction can be found in the Study Capsules.

!//(*) = 1*1(-00,00) 3 - -

ffthe set of integers .2--pts: in the interval [0, 1) i - -

|>t: (0,0) I I Ifbetween each pair of

five integers

f cally one unit at each •—o -34-&

HP describe the greatest integer function using a piecewise-definedjj|tHow does the graph of the greatest integer function differ from theIfea line with a slope of zero?

-3 -2

fSjjjjIcause of the vertical jumps described above, the greatest integer functioni|jj|li|aniple of a step function whose graph resembles a set of stairsteps. SomevDHllpf the greatest integer function are as follows.

1J = (greatest integer < - 1) = - 1

.j|ij||jo]] = (greatest integer < ^) = 0SHBBMW™""""

ons on how to usetl |3j|v5]) = (greatest integer < 1.5) = 1

d maximum l5||s;Section 1.2, you learned that a piecewise-defined function is a functioniX A; tor sp thaiteaefined by two or more equations over a specified domain. To sketch thego to this gj|g||of a piecewise-defined function, you need to sketch the graph of eachly Center. ee^^t on the appropriate portion of the domain.

Graphing a Piecewise-Defined Function

graph of f(x) =2x + 3,

-x + 4,x < 1X > 1

by hand.

Solution

ewise-defined function is composed of two linear functions. At and totof x = 1, the graph is the line given by y = 2x + 3. To the right of x = 1,jph is the line given by y = - x + 4 (see Figure 1.33). Notice that the point

l»S)Js a solid dot and the point (1, 3) is an open dot. This is because/(I) = 5.

INT Now try Exercise 43.

Most graphing utilities displaygraphs in connected mode, whichmeans that the graph has nobreaks. When you are sketchinggraphs that do have breaks, it isbetter to use dot mode. Graphthe greatest integer function[often called Int (x)] in connectedand dot modes, and compare thetwo results.

Figure 1.33


Even and Odd FunctionsA graph has symmetry with respect to the y-axis if whenever (x, y) is on the graph,so is the point ( — x, y). A graph has symmetry with respect to the origin if when-ever (x, y) is on the graph, so is the point ( — x, -y). A graph has symmetry withrespect to the x-axis if whenever (x, y) is on the graph, so is the point (x, — y). Afunction whose graph is symmetric with respect to the y-axis is an even function.A function whose graph is symmetric with respect to the origin is an oddfunction. A graph that is symmetric with respect to the x-axis is not the graphof a function (except for the graph of y = 0). These three types of symmetry areillustrated in Figure 1.34.

Symmetric to y-axisEven functionFigure 1.34

Symmetric to originOdd function

Symmetric to x-axisNot a function

Test for Even and Odd Functions

A function/is even if, for each x in the domain of /,/(—x) - f(x).A function/is odd if, for each x in the domain of/,/(—*) = —f(x).

Example 9 Testing for Evenness and Oddness

Is the function given by/(;c) = \x\, odd, or neither?

Algebraic SolutionThis function is even because

= /(*).

/CHECKPOINT Now try Exercise 59.

Graphical SolutionUse a graphing utility to enter y = \x\n the equation editor, as shown inFigure 1.35. Then graph the function using a standard viewing window, asshown in Figure 1.36. You can see that the graph appears to be symmetricabout the y-axis. So, the function is even.

10

sVfi=W?=

-10 10

-10

Figure 1.35 Figure 1.36

Section 1.3 Graphs of Functions

Example 10 Even and Odd Functions

Determine whether each function is even, odd,

a. g(x) = x3 - xb. h(x) = x2 + 1

c. f(x) = *3 - 1

Algebraic Solutiona. This function is odd because

*(--V) = (-.V)3 - (~.V)

= -Jt3 + Z

b. This function is even because

A(-JI-) = (-A-)2 + 1

37

or neither.

= x1 + 1

£ Substituting -*for* produces

/(-*) = (-.v)3 - 1

Because f(x) = x3 - 1 and -f(x) =~x*can conclude that f(-x)±f(x)

—f(x). So, the function is neither

+ 1,and

even

wn in>w, asnetric


Graphical Solution

a. In Figure 1.37, the graph is symmetric with respect to theorigin. So, this function is odd.

Figure 1.37

b. In Figure 1.38, the graph is symmetric with respect to they-axis. So, this function is even.

Figure 1.38

c. In Figure 1.39, the graph is neither symmetric with respectto the origin nor with respect to the -axis. So, this functionis neither even nor odd.

Figure 1.39

help visualize symmetry with respect to the origin, place a pin at theof a graph and rotate the graph 180°. If the result after rotation coincideste original graph, the graph is symmetric with respect to the origin


Vocabulary Check

Fill hi the blanks.

1. The graph of a function/is a collection of (x, y) such that x is in the domain of/

2. The is used to determine whether the graph of an equation is a function of y in terms of x.

3. A function/is on an interval if, for any x{ and x2 in the interval, xl < x, implies/(*,) > /(x,).

4. A function value/(a) is a relative of/if there exists an interval (*,, x-,) containing a such that

function, and is an example of a step function.

*, < * < *2 implies /(a)

5. The function /(jc) = W is called the

6. A function/is _ if, for each x in the domain off,f(—x) = f(x).

In Exercises 1- 4, use the graph of the function to find thedomain and range of/. Then find/(0).

In Exercises 5-10, use a graphing utility to graph thefunction and estimate its domain and range. Then find thedomain and range algebraically.

5. /(.x) = 2x2 + 3

6. f(x) = ~.r- - 1

7. /(A-) = Jx - 1

8. Mt) = 74 - t2

9. f(x) = \ + 3 1

In Exercises 11-14, use the given function to answer thequestions.

(a) Determine the domain of the function.

(b) Find the value(s) of x such that/U) = 0.

(c) The values of x from part (b) are referred to as whatgraphically?

(d) FJnd/(0), if possible.(e) The value from part (d) is referred to as what

graphically?

(f) What is the value of / at x = 1? What are thecoordinates of the point?

(g) What is the value of / at x = -1? What are thecoordinates of the point?

(h) The coordinates of the point on the graph of/ at whichx = -3. can be labeled (-3,/(-3)) or (-3, ).

11. f(x) = x2 - x-

13. y12. f(x) = x3 - 4x

14.

3 4

\f(x) = \x-l\-2

In Exercises 15-18, use the Vertical Line Test to determinewhether y is a function of .r. Describe how you can use agraphing utility to produce the given graph.

15. v = ir2 16. x - v2 = 16 3

-2

Section 1.3 Graphs of Equations 39

what

what

e the

e the

vhich

18. x2 = 2xy - I

In Exercises 19-22, determine the open intervals overwhich the function is increasing, decreasing, or constant.

19. /to = F 20. f(x) = x2 - 4x3

-4

21. fix) = Xs - 3x2 + 24

-5

-6

-6

-1

In Exercises 23-30, (a) use a graphing utility to graph thefunction and (b) determine the open intervals on which thefunction is increasing, decreasing, or constant.

23. /(.r) = 3

24. /(..r) = x

25. /(.r) = A-2/3

26. f(x) = -A-3/4

27. f(x) = A

28. /(.r) =

29. /(A) = \ + 11 + \ ~

30. /(.r) = - \ + 4j - \ + 11

In Exercises 31-36, use a graphing utility to approximateany relative minimum or relative maximum values of thefunction.

31. /(.r) = A2 - 6x

32. /(.r) = 3A~ - 2r - 5

33. v = 2A3 + 3A2 - lit

34. v = A3 - 6x2 + 15

35. h(x) = (x-

36. U)

In Exercises 37-42, (a) approximate the relative minimumor relative maximum values of the function by sketching itsgraph using the point-plotting method, (b) use a graphingutility to approximate any relative minimum or relativemaximum values, and (c) compare your answers from parts(a) and (b).

37. f(x) = x2 - 4x - 5

39. f(x) =x3 - 3x

41. f(x) = 3,r3 - 6x + I

38. f(x) = 3,r2 - 12*

40./Cr) = -,t3 + 3.T2

42. /to = &c - 4r2

In Exercises 43-50, sketch the graph of the piecewise-defined function by hand.

43. /W =2^ + 3, x < 0

3 - x, x > 0

, . ,, . J.T + 6, JT < -444. /to= - , ,ZT — 4, ,r > — 4

) = 7-v2 - 1 45.7

\/..'..

46.

6 47.

/W =

/to =

/•(A-) -

[A + 3, x < 03, 0 < x < 2

[2x - 1. A- > 2

( x + 5, A < -3-2.-3<x<[5x - 4, A > 1

«./'^fe:2':;;;:;50. WA-) =

,t, A < 0

1, A > 0

Library of Parent Functions In Exercises 51-56, sketchthe graph of the function by hand. Then use a graphingutility to verify the graph.

51. /W = W + 2

52. /W = W - 3

53. /(.r) = $x - \ 2

54. f(x) = f.r - 2J + 1

55. fix) = [2*1

56. /(.v) = [4rJ

In Exercises 57 and 58, use a graphing utility to graph thefunction. State the domain and range of the function.Describe the pattern of the graph.

57. ,Cr) = 2(iv - fivjj

58. gix) = 20r - Mf


In Exercises 59-66, algebraically determine whether thefunction is even, odd, or neither. Verify your answer using agraphing utility.

60. f(x) = x6 - 2x2 + 3

62. h(x) = x3 - 5

64. f(x) = xVx + 5

66. f(s) = 4s3'2

Think About It In Exercises 67-72, find the coordinates ofa second point on the graph of a function/if the given pointis on the graph and the function is (a) even and (b) odd.

67. (-1,4) 68. (-f, -7j

69. (4, 9) 70. (5, - 1)

71. (x, -v) 72. (la, 2c)

In Exercises 73-82, use a graphing utility to graph thefunction and determine whether it is even, odd, or neither.Verify your answer algebraically.

73. f ( x ) = 5 74. /(*) =-975. f(x) = 3x - 2 76. f ( x ) = 5 - 3x

77. h(x) = x2 - 4 78. f(x) = -x2 - 8

79. f(x) = 71 -x 80. g(t) = 3/t- 1

81. /(*) = \ H- 2| 82. /(*) = -1* - 5|

In Exercises 83-86, graph the function and determine theinterval(s) (if any) on the real axis for which/fr) > 0. Use agraphing utility to verify your results.

83. f(x) = 4 - x

85. f(x) = x- - 9

84. f(x) = 4x + 2

86. f(x) = x- - 4x

87. Communications The cost of using a telephone callingcard is SI.05 for the first minute and $0.38 for eachadditional minute or portion of a minute.

(a) A customer needs a model for the cost C of using thecalling card for a call lasting / minutes. Which of thefollowing is the appropriate model?

C{(t) = 1.05 + 0.38ffr - Ij

C2(t) = 1.05 - 0.38|-(r- 1)1

fb) Use a graphing utility to graph the appropriate model.Use the value feature or the zoom and trace features toestimate the cost of a call lasting 18 minutes and 45seconds.

88. Delivery Charges The cost of sending an overnightpackage from New York to Atlanta is $9.80 for a packageweighing up to but not including 1 pound and $2.50 foreach additional pound or portion of a pound. Use thegreatest integer function to create a model for the cost C ofovernight delivery of a package weighing x pounds, wherex > 0. Sketch the graph of the function.

In Exercises 89 and 90, write the height h of the rectangle asa function of x.

89. yy = —x- + 4x—l

44i --

(1,2) (3,2)

1 x 3 4 - V I 2 3 4

to

91. Population During'a 14 year period from 1990 to 2004,the population P (in thousands) of West Virginia fluctuatedaccording to the model

P = O.OlOSr4 - 0.211/3 + 0.40r2 + 7.9r + 1791,

0 < t < 14 •

where t represents the year, with / = 0 corresponding1990. i Source: U.S. Census Bureau;

(a) Use a graphing utility to graph the model over theappropriate domain.

(b) Use the graph from part (a) to determine during whichyears the population was increasing. During whichyears was the population decreasing?

(c) Approximate the maximum population between 1990and 2004.

92. Fluid Flow The intake pipe of a 100-gallon tank has aflow rate of 10 gallons per minute, and two drain pipeshave a flow rate of 5 gallons per minute each. The graphshows the volume V of fluid in the tank as a function oftime t. Determine in which pipes the fluid is flowing inspecific subintervals of the one-hour interval of time shownon the graph. (There are many correct answers.)

(60, 100)

10 20 30 40 50 60Time (in minutes)

Section 1.3 Graphs of Equations 41

angle as

) to 2004,fluctuated

1,

jonding to

1 over the

ring whiching which

iveen 1990

tank has airain pipesThe graph\inction offlowing inime shown

Synthssis

True or False? In Exercises 93 and 94, determine whetherthe statement is true or false. Justify your answer.

93. A function with a square root cannot have a domain thatis the set of all real numbers.

94. It is possible for an odd function to have the interval[0, co) as its domain.

Think About It In Exercises 95-100, match the graphof the function with the best choice that describes thesituation.

(a) The air temperature at a beach on a sunny day

(b) The height of a football kicked in a field goal attempt

(c) The number of children in a family over time

(d) The population of California as a function of time

(e) The depth of the tide at a beach over a 24-hour period

(f) The number of cupcakes on a tray at a party

95. y 96.

97. y 98. y

99. v 100.

Q»

.. ..101. Proof Prove that a function of the following form is

odd.

103. If/is an even function, determine if g is even, odd, orneither. Explain.

(a) g(x) = -/tt) (b) g(x) = /(-*)

(c)s(z) =/W -2 (A) g(x) = -f(x - 2)

104. Think About It Does the graph in Exercise 16 representx as a function of yl Explain.

105. Think About It Does the graph in Exercise 17represent x as a function of yl Explain.

106. Writing Write a short paragraph describing threedifferent functions that represent the behaviors ofquantities between 1995 and 2006. Describe one quantitythat decreased during this time, one that increased, andone that was constant. Present your results graphically.

Skills Review

In Exercises 107-110, identify the terms. Then identify thecoefficients of the variable terms of the expression.

107. -2X2 +

109. - 5x2

108. 10 + 3x

110. lx4 + V

In Exercises 111-114, find (a) the distance between the twopoints and (b) the midpoint of the line segment joining thepoints.

111. (-2,7), (6,3)

112. (-5,0), (3, 6)

113. (f,-l),H, 4)

114- (-6, f), (i |)

In Exercises 115-118, evaluate the function at each specifiedvalue of the independent variable and simplify.

115. f(x) = 5x - 1

116. f(x) = -x2 - x + 3

117. f(x) = x-JlT^l

(a) /(3) (b) /(12)

118. /W = -kxx + 1

(c) /(6)

f In Exercises 119 and 120, find the difference quotient andsimplify your answer.

102. Proof Prove that a function of the following form is 120. f(x) = 5 + 6x - x2, + h) ~ /(6)even.


Vocabulary Check

In Exercises 1-5, fill in the blanks.

1. The graph of a is U-shaped.

2. The graph of an is V-shaped.

3. Horizontal shifts, vertical shifts, and reflections are called.

., while a reflection4. A reflection in the ;t-axis ofy= f(x) is represented by h(x) -in the y-axis o f y = f(x) is represented by h(x) = .

5. A nonrigid transformation ofy = f(x) represented by cf(x) is a vertical stretch ifa vertical shrink if .

6. Match the rigid transformation of y = f(x) with the correct representation, where c > 0.

(a) h(x) = f(x) + c (i) horizontal shift c units to the left

(b) h(x) = f(x) - c (ii) vertical shift c units upward

(c) h(x) = f(x - c) (in) horizontal shift c units to the right

(d) h(x) =f(x + c) (iv) vertical shift c units downward

and

In Exercises 1-12, sketch the graphs of the three functionsby hand on the same rectangular coordinate system. Verifyyour result with a graphing utility,

2. /W = F

g(x) = kx + 2

h(x) = \(x - 2)

4. /W = x2

g(x) = x2-4

h(x) = (x + 2)- + 1

6. /W = (x - 2)2

g(x) = (x + 2)- + 2

AW = ~(x- 2)- - 1

8. /W = x-

g(x) = \x + 2

1. fix) = x

g(x) = x-4

h(x) = 3*

3. f(x) = x-

g(x) = x- + 2

h(x) = Cr - 2j2

5. f(x) = -x2

g(x) = -x- + 1

h(x) = -Ct-2,F

7. f(x) = x^

h(x) = (2xY-

9. /W = W

g(x\ |*| - 1

h(x) = \x-3\. f(x) = v'1

g(x] = -Jx +

10.x+3

h(x) = -2 - 1

12.

13. Use the graph of / to sketch each graph. To print anenlarged copy of the graph, go to the websitewww. mathgraphs. com.

(a) y=f(x) +2

(b) y = -/W

(c) y=f(x-2)

(e) y = 2f(x)

(f) y=/Kv)(g)y=f(ix) ' -3-f-CO,-!)

14. Use the graph of / to sketch each graph. To print anenlarged copy of the graph, go to the websitewww.rnathgTaphs.com.

(a) y=f(x) - 1

(b)y=/(,t+ 1)

(c) y=f(x- I)

(d) y = -f(x - 2)

(e) y=f(-x)

(f) v = i/W(g) y=f(2x)

(-2.4)

(3--D

print anwebsite

print anwebsite

In Exercises 15-20, identify thdescribe the transformation show!equation for the graphed function.

fecom ,.4 swttt,,. Reflect™,, and SMching topte

ction and di «^i _ .•?

15.

-3

16.

-7

17.

-3 \ /y18.

27' -' - I' + 3"pare ae •-» »f *•

, - w - 3

39.

= f(x + 2)

= I/W

40.

=/(*- i)

49

2. /(^) = *3 - 3,r2 + 2sw = -f/W #W = -/toAW =/(-*) *W=/(2r)

In Exercises 43-56, £ is related to one of the six parentfunctions on page 42. (a) Identify the parent function/, (b)Describe the sequence of transformations from / to g. (c)Sketch the graph ofg- by hand, (d) Use function notation towrite g in terms of the parent function/.*" i •>43. g(x) = 2 - (x + 5)2

45. g(x) = 3 + 2(x - 4)2

47. g(x) = 3(jf - 2)3

49. sfr) = (* - 1)3 + 2

51. sW = \ + 4[ + 8

53. g(x) = -2\x-55.

- 4

44. £to = ~(x+ 10)2 + 5

46. gto = -jU + 2)2 - 2

48. 0^W = -k(x + I)3

50. g(x) = -(.t + 3)3 - 10

52. sto = \x + 3| + 9

54. g(x) = ||x - 2| - 3

56. g(x) = - VJT + 1 - 6

57. F«e/ Use The amounts of fuel F (in billions of gallons)used by vans, pickups, and SUVs (sport utility vehicles)from 1990 through 2003 are shown in the table. Amodel for the data can be approximated by the functionF(t) = 33.0 + 6.2V?, where t = 0 represents 1990.(Source: U.S. Federal Highwa;.- Administration,

Year Annual fuel use, F(in billions of gallons)

1991

1992

1993

1994

1995

1996

1997

1998

1999 I

2000

2001

2002

38.2

40.9

42.9I 44.1

45.6

47.4

49.4

50.5

52.8

52.9

53.5

55.22003 ,, „1 OQ.J

(a) Describee transformation of the parent funcdon

' i The3 IT111112 Utility l° Sraph *e m°del and *« datam the same viewing window.(O

Explain how you got your answer.


58. Finance The amounts M (in billions of dollars) of homemortgage debt outstanding in the United States from 1990through 2004 can be approximated by the function

M(t) = 32.3f2 + 3769

where t—0 represents 1990.

Library of Pareni Funciions In Exercises 65-68,determine which eqoation(s) may be represented by thegraph shown. There may be more than one correct answer.

(a) Describe the transformation of the parent function/(/) = t2-

(b) Use a graphing utility to graph the model over theinterval 0 < t < 14.

(c) According to the model, when will the amount of debtexceed 10 trillion dollars?

(d) Rewrite the function so that t = 0 represents 2000.Explain how you got your answer.

Synthesis


59. The graph of y = j{ - x) is a reflection of the graph ofy = fix) in the A--axis.

60. The graph of .v = —f(x) is a reflection of the graph ofv = fix) in the y-axis.

61. Exploration Use the fact that the graph of y = fix) has^-intercepts at x = 2 and x = - 3 to find the .r-intercepts ofthe given graph. If not possible, state the reason.

(a) y=fi-x) (b) y = -fix) (c) y = 2fix)

(d) v = fix) + 2 (e) y = fix - 3)

62. Exploration Use the fact that the graph of y = fix) has^-intercepts at;c = - 1 and x = 4 to find the z-intercepts ofthe given graph. If not possible, state the reason.

i&)y=fi-x) (b)y=-/0r) (c)y = 2fix)

(d )y= /0 r ) - l it) y= fix -2)

63. Exploration Use the fact that the graph of y = fix) isincreasing on the interval (—00, 2) and decreasing on theinterval (2, co) to find the intervals on which the graph isincreasing and decreasing. If not possible, state the reason.

(a) y = fi~x) (b) y = -fix) (c) y = 2fix)

(d) y =fix) - 3 (e) y = fix + 1)

64. Exploration Use the fact that the graph of y = fix) isincreasing on the intervals (-00, -1) and (2, co.) anddecreasing on the interval (—1.2) to find the intervals onwhich the graph is increasing and decreasing. If notpossible, state the reason.

(a) y = fi-x) (b) y = -fix) (c) y = kfix)

(d) y = -fix - 1) (e) y = fix - 2) + I

(a) f(x) = \ + 2 -M

(b) f(x) = \ - 1 + 2

(c) /(.t) = \x - 2\ 1

(d) f(x) = 2 + x-1\) /(.t) = | Or-2) + I

(f) f(x) = 1 - \x - 2

67. y 68.

(a) /Or) =

(b) /Or) =

(0 f(x) =(d) f(x) =

(e) fix) =

(f) fix) =

(a) fix) = ix - 2F - 2

(b) fix) = ix + 4j2 - 4

(c) fix) = CT - 2)= - 4

(d) fix) = Or + 2); - 4

(e) fix) = 4 - Or - 2)2

(f) /Or) = 4 - Or + 2V-

Skiiis Review

(a) fix) = - ix - 4Y + 2

(b) fix) = - ix + 4)J + 2

(c) fix) = -ix -2)3 - 4

(d) /Or) = i-x - 4)3 + 2

(e) fix) = Or + 4)3 + 2

(f) /Or) = i-x + 4)3 + 2

In Exercises 69 and 70, determine whether the lines ij andL., passing through the pairs of points are parallei. perpen-dicular, or neither.

69. L,: (-2, -2), (2, 10)

L,: (-1,3), (3, 9)

70. I,: (-1. -7), (4,3) -

L,: (1.5), (-2, -7)

In Exercises 71-74, find the domain of the function.

4


Vocabulary Check

Fill in the blanks.

1. Two functions/and g can be combined by the arithmetic operations of , .and to create new functions.

2. The of the function/with the function g is ( / » g ) ( x ) = f(g(x)).

3. The domain of / ° g is the set of all x in the domain of g such that is in the domain off.

4. To decompose a composite function, look for an and an function.

In Exercises 1-4, use the graphs of / and g to graphh(x) = (/ + g)(x). To print an enlarged copy of the graph,go to the website tvww.mathgraphs.com.

4.

In Exercises 5-12, find (a) (/ + g)(*), (b) (f - g)(x\) (/?)(*), and (d) (flg)(x). What is the domain of

5. f(x) = x + 3, g(x) = x-3

6. f(x) = 2x-5, g« = 1 -

7. /M = A--, g« = l-x

8. /(x) = 2x - 5. g(x) = 4

9. /U) = *2 + 5, glx) = v7

10. f(x) = Vx2 - 4, gU) =

11. /(x) = -. g(x) = ~X X~

g(x) =

In Exercises 13-26, evaluate the indicated function forf(x) = x2 - 1 and g(x) = x - 2 algebraically. If possible,use a graphing utility to verify your answer.

i3.(f+g)(3) 14. (/-*)(-2)

16. (/+*)(!)

18. (/«)(-6)

%20.

22. ( / + g ) ( r - 4 )

24. (/g)(3f2)

26. (I

19. (A-5)Vg/

21. (/-g)(2/)

23. (fg)(-5t)

A-.)

In Exercises 27-30, use a graphing utility to graph the func-tions/, g, and h in the same viewing window.

27. f(x) = {x, g(x)=x-\. h(x] = f(x) + g(x)

28. f(x) = \x, g(x) = -x + 4, h(x) = f(x) - g(x)

29. f ( x ) = x-, g(x) = -2x, h(x) =f(x) • g(x)

30. f(x) = 4 - x\ = x, h(x) = /W/g(.t)

la Exercises 31-34, use a graphing utility to graph/, g, and/ + g in the same viewing window. Which function con-tributes most to the magnitude of the sum when 0 < x <• 2?Which function contributes most to the magnitude of thesum when x > 6?

31. /(.T) = 3.x. g(x) = -'—

32. f(x) = ^ g(x) = V*

33. f(x) = 3,v + 2, gU) = - •

34. /( .r)=.x2-i gW = -3,

the

53.

i fea-sible,

fuec-

f, andi con-

2?the

In Exercises 35-38, Snd (a) / ° g, (b) g »/, and, if possible,

38. f(x) =

IB Exercises 39-48, determine the domains of (a) /,(b) g, and (c) / ° g. Use a graphing utility to verify yourresults.

39. f(x) = Vx + 4, g(x) = x2

40. fix) = V',r + 3, g(*) = |

41. /tr) = -t2 + 1 , gCt) =

42. /W = x{'4, g(x) = r*

43. fix) = -, g(x) = x+3

44. fix) = p g« =

45. - 4| , glx) = 3 - x

46. fix) = - r , gix) = x - 1

47.

48.

1- 4

= X + 1

In Exercises 49-54, (a) Snd f°g, g »/, and the domain of/»g. (b) Use a graphing utility to graph f°g and g «/.Determine whether f°g = g ° f -

49. /M = 7,r + 4, gi.-r) = ^:2

50. f i x ) = + 1 , g(x) = x3 - 1

51. /Cv) = \x - 3, g(:c) = 3;c + 9

52. /(jt) = J~x, g(x) = V^

53. fix) = ,x-2/5, gU-) = A-6

54. /U) = \ , g(x) = -x2 + 1

In Exercises 55-60, (a) find (f°g)(x) and (g °f)(x), (b)determine algebraically whether (f°g)(x) = (g °f)(x). and(c) use a graphing utility to complete a table of values forthe two compositions to confirm your answers to part (b).

55. fix) = 5* 4- 4, g(x) = 4 - x56. f(x) = ICT - i). £(*) = 4,-c + 1

57- /W = V-t + 6, gC58. (A- = jJ _

Section 1.5 Combinations of Functions

X)=\X, gix) = 2AT3

60.

In Exercises 61-64, use the graphs of / and g to evaluate thefunctions.

1 2

61. (a) (/ + g)(3)

62. (a) (/-g)(l)

63. (a) (/-$)(2)

64. (a) (/°g)(l)

(b)

(b)

(b) (g «/)(2)

(b) (g °/)(3)

f In Exercises 65-72, find two functions / and g such that(/ ° g)W = ^(-^)- (There are many correct answers.)

65. hix) = (2x+ I)2

67. hix) = 3/x- - 4

69.

66. hix) = (1 - x)

68. /zCv) = 79 -

70.

. 2

42J2

71. h(x) = (x + 4)- + 2(x + 4)

72. h(x) = (x + 3)3/2 + 4(x + 3)1/2

73. Stopping Distance The research and developmentdepartment of an automobile manufacturer has determinedthat when required to stop quickly to avoid an accident, thedistance (in feet) a car travels during the driver's reactiontime is given by

10

where x is the speed of the car in miles per hour. Thedistance (in feet) traveled while the driver is braking isgiven by

Bix) = ±x\) Find the function that represents the total stopping

distance T.

(b) Use a graphing utility to graph the functions R, B, andT in the same viewing window for 0 < x < 60.

(c) Which function contributes most to the magnitude ofthe sum at higher speeds? Explain.


74. Sales From 2000 to 2006, the sales £, (in thousands ofdollars) for one of two restaurants owned by the same par-ent company can be modeled by R{ = 480 - 8r - 0.8r2,for t = 0,1, 2, 3, 4, 5, 6, where t = 0 represents 2000.During the same seven-year period, the sales R2 (in thou-sands of dollars) for the second restaurant can be modeledby R2 = 254 + 0.78r, for t = 0, 1, 2, 3, 4, 5, 6.

(a) Write a function R3 that represents the total sales for thetwo restaurants.

(b) Use a graphing utility to graph R^R-,, and R3 (the totalsales function) in the same viewing window.

Data Analysis In Exercises 75 and 76, use the table, whichshows the total amounts spent (in billions of dollars) onhealth services and supplies in the United States and PuertoRico for the years 1995 through 2005. The variables y^y2,and j>3 represent out-of-pocket payments, insurance premi-ums, and other types of payments, respectively. (Source:

*- Year

1995

19961997

1998

1999

2000

2001

2002

20032004

2005

y\6

152

162

176

185

193

202

214

231

246

262

2

330

344

361

385

414

451

497

550

601

647

691

y?

457

483

503

520

550

592

655

718

766

824

891

The models for this data are y1 = 11.4* + 83,j>2 = 2.31*2 - 8.4* + 310, and y3 = 3.03*2 - 16.8* + 467,where * represents the year, with t = 5 corresponding to1995.

75. Use the models and the table feature of a graphing utility tocreate a table showing the values of y\, v-,, and y^ for eachyear from 1995 to 2005. Compare these values with theoriginal data. Are the models a good fit? Explain.

76. Use a graphing utility to graph y t , y-,, y3, andyT = y | + v-> + y3 in the same viewing window. Whatdoes the function yT represent? Explain.

77. Ripples A pebble is dropped into a calm pond, causingripples in the form of concentric circles. The radius (in feet)of the outermost ripple is given by r(t) = 0.6f, where t isthe time (in seconds) after the pebble strikes the water. Thearea of the circle is given by A(r) = irr2. Find and interpret(A ' r)(r).

78. Geometry A square concrete foundation was prepared asa base for a large cylindrical gasoline tank (see figure).

(a) Write the radius r of the tank as a function of the lengthx of the sides of the square.

(b) Write the area A of the circular base of the tank as afunction of the radius r.

(c) Find and interpret (A = r)(x}.

79. Cos* The weekly cost C of producing x units in a manu-facturing process is given by

C(x) = 6(k + 750.

The number of units x produced in t hours is x(t) = 50r.

(a) Find and interpret C(x(t)).(b) Find the number of units produced in 4 hours.

(c) Use a graphing utility to graph the cost as a function oftime. Use the trace feature to estimate (to two-decimal-place accuracy) the time that must elapse until the costincreases to $15,000.

80. Air Traffic Control An air traffic controller spots twoplanes at the same altitude flying toward each other. Theirflight paths form a right angle at point P. One plane is 150miles from point P and is moving at 450 miles per hour.The other plane is 200 miles from point P and is moving at450 miles per hour. Write the distance s between the planesas a function of time t.

r= 200-f:s

100--

100 200

Distance (in miles)

81. Bapr<1 .deen

(b

(c

82. Piciccth

83. Sn

84.

Tnuthe

Section 1.5 Combinations of Functions 61

gl. Bacteria The number of bacteria in a refrigerated foodproduct is given by N(T) = IOT2 - 207 + 600, for1 < T £ 20, where T is the temperature of the food indegrees Celsius. When the food is removed from the refrig-erator, the temperature of the food is given byT(t) = It + 1, where t is the time in hours.

(a) Find the composite function N(T(f)) or (N ° T)(t) andinterpret its meaning in the context of the situation.

(b) Find (N ° T)(6) and interpret its meaning.

(c) Find the time when the bacteria count reaches 800.

82. Pollution The spread of a contaminant is increasing in acircular pattern on the surface of a lake. The radius of thecontaminant can be modeled by r(t) = 5.25 Vr, where r isthe radius in meters and t is time in hours since contamina-tion.

(a) Find a function that gives the area A of the circular leakin terms of the time t since the spread began.

(b) Find the size of the contaminated area after 36 hours.

(c) Find when the size of the contaminated area is 6250square meters.

83. Salary You are a sales representative for an automobilemanufacturer. You are paid an annual salary plus a bonus of3% of your sales over $500,000. Consider the two func-tions

f(x) = x- 500,000 and g(x) = 0.03,x.

If x is greater than $500,000, which of the following repre-sents your bonus? Explain.

(a) f(g(x)) (b) g(/W)

84. Consumer Awareness The suggested retail price of a newcar is p dollars. The dealership advertised a factory rebateof $1200 and an 8% discount.

(a) Write a function R in terms of p giving the cost of thecar after receiving the rebate from the factory.

(b) Write a function S in terms of p giving the cost of thecar after receiving the dealership discount.

(c) Form the composite functions (R ° S)(p) and (5 ° R)(p)and interpret each.

(d) Find (R » 5)( 18,400} and (S ° R)( 18,400). Which yieldsthe lower cost for the car? Explain.

Synthesis


85. If f(x) = x + i and g(x) = 6x, then

86. If you are given two functions f(x) and g(x), you cancalculate (/ ° g)(x) if and. only if the range of g is a subsetof the domain off.

Exploration In Exercises 87 and 88, three siblings are ofthree different ages. The oldest is twice the age of the mid-dle sibling, and the middle sibling is six years older thanone-half the age of the youngest.

87. (a) Write a composite function that gives the oldest sib-ling's age in terms of the youngest. Explain how youarrived at your answer.

(b) If the oldest sibling is 16 years old, find the ages of theother two siblings.

88. (a) Write a composite function that gives the youngestsibling's age in terms of the oldest. Explain how youarrived at your answer.

(b) If the youngest sibling is two years old, find the ages ofthe other two siblings.

89. Proof Prove that the product of two odd functions is aneven function, and that the product of two even functions isan even function.

90. Conjecture Use examples to hypothesize whether theproduct of an odd function and an even function is even orodd. Then prove your hypothesis.

91. Proof Given a function /, prove that g(x) is even andh(x) is odd, where g(x) = ^[f(x) + f(-x)] andh(x) = 2if(x) -/(-*)].

92. (a) Use the result of Exercise 91 to prove that any functioncan be written as a sum of even and odd functions.(Hint: Add the two equations in Exercise 91.)

(b) Use the result of part (a) to write each function as asum of even and odd functions.

1

~~ ° x + 1

Skills Review

In Exercises 93-96, find three points that lie on the graph ofthe equation. (There are many correct answers.)

94. y = j.T3 - 4x- + 193. y = — x2 + x — 5

95. y? + y- = 24 96. y = -x- — 5

In Exercises 97-100, find am equation of the line that passesthrough the two points.

97. (-4,-2), (-3, 8) 98. (1,5), (-8, 2)

99. (!. - l), (-1, 4) 100. (0.1.1), (-4. 3.1)

Section 1.6 Inverse Functions 69

Vocabulary Check

Fill in the blanks.

- 1. If the composite functions /(^and is denoted by _ _ .

2. The domain of/is the

.*)) - x, then the function g is the . function of/

2.

4

5. /W = IT + 1

7. /W =

6- /to =

8- /to =

= - r3*

= *-3

- 1

In Exercises 9-14, (a; show that f and a * -

9. /to = - , -3,

11.

12.

5, g(x) =

the graphs eSC"be the rela^Wp between

18. /W = 9-x2, x>0; g(x) =

19. f(x) = 1 - x3, gix) = T^l

20. fix) = — -^, z > 0; gix) = -^i, 0 < x < I1 ~f~ ^v

In Exercises 21-24, match the graph of the function withthe graph of its inverse function. [The graphs of the Inversefunctions are labeled (a), (b), (c), and (d).]

(a)

-3

-1

(b)

9 -3

-1

(C)

-6

21.

-6

(d)

e -s

">i

-4-3

\1

23.

-3

-1

24.

-6

70Chapter I Functions and Their Graphs

25./(A-) = 2*, g(x) = ±

26. /(.r) = x - 5, g(x) =x + 5

27-/« = ~j *W--£±-IX -r 3 X — I

x — >." "' ' x — 1

In Exercises 29-34, determine if the graph is that of a func-tion. If so, determine if the function is one-to-one.

30. v

32.

34.

In Exercises 35-46, use a graphing utility to graph thefunction and use the Horizontal Line Test to determinewhether the function is one-to-one and so an inversefunction exists.

35. /to = 3 - |,

37. h(x] = - -T' -x- + 1

39. M.v) = VT6 - ,

41. /Ct) = 10

J3- o W = (A- -f- 5)3

44. ,c) = _

36. /to = i(,v + 2 ) 2 -

4 - A-38. *(A-) =6,v2

42./W = -0.65

45. 4 - |* - 4

In Exercises 47-58, determine algebraically whether thefunction is one-to-one. Verify your answer graphically.

47. /to = x4

... ,, ,49. /(A-) =

51. /(,r) = -1x-

53. /(.r) = (x+ 3)2,

54. q(x) = (x - 5)2,

55. /(A-) = v/lTTT

56. /W = yT l57. /(A-) = |A- - 2|,

48. g(x) =X2_ X4

50./ft = 3*+ 5

52. h(x) = -i

> -3

< 5

< 2

58. —x2 + I

In Exercises 59-68, find the inverse function of /algebraically. Use a graphing utility to graph both/and/~!

in the same viewing window. Describe the relationshipbetween the graphs.

59. /(,r) = IY - 3

61. /(A-) = xs

63. /to = .r3''5

65. /(.T) = 74=1

60. /(A-) = 3.t

62. /to = A-3 •

64. /(.r) = A2 ,0 < A- < 2

> o

66. /to = Vl6 - A-2, -4 < A- < 0

67. /(A-) =x 68. /U) = - ,

Think About It In Exercises 69-78, restrict the domain ofthe function/so that the function is one-to-one and has aninverse function. Then find the inverse function/"1. Statethe domains and ranges of/and/"1. Explain your results.(There are many correct answers.)

69. /(.T) = (x - 2)2

70. /W = 1 - x±

71. /W = |A- + 21

72. /to = JA- - 21

73. /to = (x + 3)2

74. /to = (.r - 4)2

75./U) = -2A-- + 5

76. /U) = it2 - 1

77. /Cv) = |.T - 4| + 1

78. /(.T) = - |.Y - 11 - 2

Irto

C-rgr;«vth«invrea

89.

91.

Section 1.6 Inverse Functions 71

her the

>n of /'and/-1

.tionship

: 0

>main ofd has an-1. State• results.

In Exercises "complete the

79- ]

4-

2-

"'A«•--

19 and 80, use thetable and sketch the

ft,,.- 2 4

graph of the function / to 95. (/~ ' °/- ')(6) 96. (g~ ' °g~ ')( - 4)

grapoof/-'. 97.tf.gr1 »,-'•/-

x

-4

-2

1

3

/-'to In Exercises 99-102, use the. functions /(r) = x + 4 andg(x) = 2x - 5 to find the specified function.

99. £-'°/~' 100. /- '«£-'

101. (/•*)" 102. (*•/)-'

103. SA0£ Sizes The table shows men's shoe sizes in theUnited States and the corresponding European shoe sizes.

In Exercises 81-88, use the graphs of_y =f(x) and y = g(x)to evaluate the function.

87. (j ./-'){2)

Graphical Reasoning In Exercises 89-92, (a) use agraphing utility to graph the function, (b) use the drawinverse feature of the graphing utility to draw the inverse ofthe function, and (c) determine whether the graph of theinverse relation is an inverse function, explaining yourreasoning.

89. f(x) = ,r3 + x + 1

91. g(x) =x-+ 1

92. /(.t) =

In Exercises 93-98, use the functions /(i) = ±x - 3 and?(*) = x3 to find the indicated value or function.

93- Cr'^'Xl) 94. (e- 'of-

Let y - f(x) represent the function that gives the men'sEuropean shoe size in terms of x, the men's U.S. size.

3 Men's U.S.shoe size

8910111213 .

Men's Europeanshoe size

414243454647

(a j Is/one-to-one? Explain.(b) Find/(ll).(c) Find/"'(43), if possible. f-.

(d) Find/(/-'(41)). "'^ ", • - - 7 \j Find/"'(/(13)). VP >

104. 5Aoe S/zss The table shows women's shoe sizes in thef^yUnited States-and the corresponding European shoe sizes. /• £Let_y = g(,v) represent the function that gives the women's '„(/European shoe size in terms of .t, the women's U.S. size.

Women's U.S.* shoe size

4

5

6

7

8o

Women's Europeanshoe size

353738394042

(a) Is g one-to-one? Explain.

(b) Find #(6).(c) Find f1 (42).(d)

(e)


105. Transportation The total values of new car sales / (inbillions of dollars) in the United States from 1995 through2004 are shown in the table. The time (in years) is givenby t, with r = 5 corresponding to 1995. (Source:National Automobile Dealers Association;

~BiJ«-.', y~:

Year, t

561891011121314

Sales,/®

456.2490.0507.5546.3606.5650.3690.4

679.5699.2714.3

(a) Does/ ' exist?

(b) If/"' exists, what does it mean in the context of theproblem?

(c) If/-' exists, find/-'(650.3).

(d) If the table above were extended to 2005 and if thetotal value of new car sales for that year were $690.4billion, would/'1 exist? Explain.

106. Hourly Wage Your wage is $8.00 per hour plus $0.75for each unit produced per hour. So, your hourly wage vin terms of the number of units produced x is>• = 8 + 0.75.x.

(a) Find the inverse function. What does each variable inthe inverse function represent?

(b) Use a graphing utility to graph the function and itsinverse function.

(c) Use the trace feature of a graphing utility to find thehourly wage when 10 units are produced per hour.

(d) Use the trace feature of a graphing utility to find thenumber of units produced when your hourly wage is$22.25.

Synthesis

True or False? In Exercises 107 and 108, determinewhether the statement is true or false. Justify your answer.

107. If/is an even function,/"1 exists.

108. If the inverse function of/exists, and the graph of/has av-intercept, the -intercept of/is an -intercept of/~'.

109. Frvof Prove that if / and g are one-to-one functions.

110. Proof Prove that if/is a one-to-one odd function,/-1 ]s

an odd function.

In Exercises 111-114, decide whether the two functionsshown in the graph appear to be inverse functions of eachother. Explain your reasoning.

111. 112.

113. 114.

T

In Exercises 115-118, determine if the situation could berepresented by a one-to-one function. If so, write a state-ment that describes the inverse function.

115. The number of miles n a marathon runner has completedin terms of the time t in hours

116. The population p of South Carolina in terms of the year •from 1960 to 2005

117. The depth of the tide d at a beach in terms of the time .over a 24-hour period

118. The height h in inches of a human born in the year 2000in terms of his or her age n in years

Skills Review

In Exercises 119-122, write the rational expression insimplest form.

120.

122.121. 6 - x x- - It - 10

In Exercises 123-128, determine whether the equationrepresents y as a function of AT.

123. 4x - v = 3 124. x = 5

125. x2 + r = 9 126. x- + y = 8

127. v = Jx + 2 128. x - y- = 0

Chapter 1 Functions and Their Graphs

1.7 Exercises

Vocabulary Check

Fill in the blanks.

2. Consider a collection of ordered pairs of the form (.v. vl. II v tends to decrease as A increasethen the collection is said to have a correlation.

3. The process of finding a linear model for a set of data is called .

•4. Correlation coefficients vary between and ______ .

( 1 . 5 , 4 1 . 7 ) , 11 .0 ,32 .4) , (0 .3 ,19 .2 ) , (3.0. 48.4), (4 . ( i . :(0.5.28.5!. (2.5,50.4). (1 .8 ,35 .5) , (2.0, 36.0),(1 .5 ,40.0) , 13.5,50.3). (4.0. 55.2). (0.5. 29 ,1) . (2.2.-(2.0. 4 L6J

( a ) Create a scalier plot of the data,

( h i Does the relat ionship between .v and v appealapproximately linear? E x p l a i n .

( a j Create a scatter plot of the data.

(b) Does the re la t ionsh ip between consecutive L|ui/ . scoresappear to be approximately linear.1 I I not , give somepossible explanations.

In Kxercises3-(i, the scatter plots of sets of data are shown.Determine whether there is positive correlation, negativecorrelation, or no discernible correlation between thevariables.

3. * 4. '•

In Exercises 7-10, (a) for the data points given, draw a lineof best fit through two of the points and find the equation ofthe line through the points, (b) use the regression feature ofa graphing uti l i ty to find a linear model for the data, and toidentify the correlation coefficient, (c) graph the data pointsand the lines obtained in parts (a) and (b) in the sameviewing window, and (d) comment on the validity of bothmodels. To print an enlarged copy of the graph, go to thewebsite www.mathgraphs.com.

Section 1.7 Linear Models and Scatter Pints

4(0. 7)

muoke'x Law Hooke's Lav. suites thai thered tn compress nr stretch a spring ( w i t h i n i

; - , h i , t s ! is proportional lo the d is tance il that the„•. .i-pressed or stretched from ils original length/- \d. where /, is the measure of the s t i l l n e- p / i n g and is cal led the xprhi^ n'uMtmi. The la:hj elongation J in centimeters o! a spring when,'- K i log rams is appl ied.

i c . L 'se the >'i'xn-\\ion feature ot a graphing u t i l i l 1

a l inear model for the data. Compare ihK ir.othe model from part ( h ) .

i d : L'se the model from part i c i (o estimate the cuof the spring when a force of 55 k i l o g r a m s i > ;

\2. Cell Plumes The average [en^ths L of ce lk i i acal ls in minu tes from 1999 to 2004 are shown in r

j Average length, LI (in minutes}

Use a graphing u t i l i t y to create a scatter pidata, u i lh ? = 9 corresponding to 1999.

' Use the H'ltri'.'ixioii feature of a graphing u l i i ia l i n e a r model lor the data. Let t representv > i t h i - 9 corresponding to 1999.

1 Lse a graphing u t i l i t y to plof the data and ;model in the same v i e u i n g window. Is thegood fit1.1 Explain.

) Lse the model to predict the average lengths iphone calls for the years 2010 and 2015.answer seem reasonable" Expla in .

Mean salary. Sfin thousands of dollars)

(a i I ' S L i ji i p h m _ u h l i l \o ere ikdat i suih i = 0 LOiaspond i i i ' t i i

( b i IV the iLi,n \i n tc mi a it t _ i \'a l inea r model for the data. Let iw i t h ! - 0 corresponding to 2 ( H ) H

i c ) Use a g raph ing u i i l i i s ' ploi t i nmodel in the san.e vicwini: \ \ i n d tgood l i t I \pl nn

u l i Use the model to pred ic t UIL iprofession il footbal l nl i \ u s m 2l i^resul ts set_m re ison ihle ' F\pl in

ic) Wh il i the slope, nt \u i i i mode \about the mean salaries ol pplayers1.1

!4. Teacher's Salaries The nx in s il idol lars) of p u b l i c school leadurs m h '!<W to 2004 are shoni L

Mean salary. 5(in thousands of dolirs-

L'se a graphing u t i l i t y to create adara . \ \ i t h / = 9 corresponding to i

i L'se the ivsraixion i 'ealure of a giaa l inear model for the data. Lc: ;u i t h / - ll con'esponding lo 199L i .

L'sc a graphing u t i l i t y to plot themodel in the same viewing w i n dgood lit'.' Explain,

i Use the model lo predict t t ; ni.anin 2005 and 2010. Do tru ''c •P:\plain.

u i j Use LI g raphing u t i l i t s to create a scaiie: pi in .Jata. w i t h ; - 5 corresponding to 2005.

( h i Use (he m,'/-<'w/i;/! feature of a i j raphum u ; i h l ndincar model lor the data . Lot i represen

\ \ i t h l - 5 corresponding io 2005.

( c i Use a graphing u t i l i t y to p lo t tlie daia .md L. [ h ' l iemode! :n (lie same \ ie \^ \ \ indow. N '.he M agood tit '? Exp la in .

id) Use the model lo predicl the population or N L Jerse\n 2050. Does the result seem reasonable': h\ in

17. State Population The projected popub t io i i / inthousands! far selected years tor Wyoming iv i -^eL > i the20110 census are shown in ihe table " ' -

YearPopulation, P(in thousands}

• . u i Use a graphing u l i l i l v lo create a scalier pio; oi thedala. wi th i = 0 corresponding to 1990.

• h i Use the rc^rvwo/i f ea tu i e of a graphing mi lk} so f i n da l inear model lor the data and to idemifs :hecorrelation coeff icient . Let ' represent the \ear , u i t h.' — (1 corresponding to ] 9 ' J O .

; c : Graph the model wi th the data in the same v i e w i n gwindow.

; d > Is the model a good fit for the data'.1 E x p l a i n

i e ! Use the model lo predict the average montlm cable. b i l l s for the years 2005 and 2010.

i f ; Do you believe ihe model would be accurate ;o predictthe average month 1\e h i l l s for f u t u r e years?Explain .

1ft . State Population The pro jec ted p o p u l a t i o n s f i i nt housands ) for selected years lor New .lersey based on the2000 census are shown in the t ab l e . ^ • •. . ' •

un Use a graphing u t i l i t y to create a scatter pk i \edata, \\itii .' = 5 corresponding to 2005.

( h ) Use the >Ti>nj.\-xion feature of a uraphing u n l i t o Imda l inea r model for the data. Lei i represent the \car 'wi th ,' = 5 corresponding lo 2005.

(o Use a graphing u t i l i t y to plot the data and g iph ihe -model i;i the .same \ i e w i t i i 2 w indow. Is the K dd agood I'll? E x p l a i n .

i d ) Use the model to predict ihe populat ion ot \Vu mn in2050. Docs Ihe resul t seem reasonable1.1 H\p!a n

IS . Advertising and Sales The (able shows the a d \ i _ i [ismsexpendi tures v and sales volumes v for a company i se\enrandoml} selected months . Both arc measured in t i n us mdsof dollar-,.

• , i ; l.'se the model to e s t ima te sales j" iexpendi tures of SI5011.

1'). \nmber of Stores The table shows the number s T otTarget stores from 1997 lo 2 ( i ( ) h . -, • • . . < .

Year

j 1997

i 1998

\9

2000

; 2001

j "HK'P

2003

2004

2005

2006

Number of stores, T

i ISc ihe rt'ffre.'ixion feature o! a graphing u t i l i ty to find.1 l i n e a r model for the data and to i d e n t i f y thecorrelation coefficient. Let ; represent ihe \ear, wi thi - 7 corresponding to 1997.

! L ' s e a graphing u t i l i t y to plot the data and graph the:rni iel in the same viewing window.

YearPopulation, jP(in thousands)

Month

I

Advertisingexpenditures, .v

2.4

2 1 . 6

5 2.0

4 ; 2.6

5 ' 1 .4

(.-

1.6

2.0

Salesvolume, v

202

184

220

240

I S O

!64

L_ l_^J

Synthesis

Trite or False? In Exercises 21 and 22, cietermine \vhetlierthe statement is irue or false. Justify \our ans\\er.

21. A l inear regression model w i t l i ,1 p o M t i \ cor re la t ion \ \ i l iha \ a slope t h a t is greater than 0.

22. If the correlation coefficient fo ra l i n e a l regression model isclose to -• I . ihe regressi--" l i n e canno: S^e used to describe[lie data.

23. Writing A l inear mathemat ica l model lor p red ic t ing pr i /ewinn ings al a race is based on data (or ? years. Write aparagraph discussing the potent ia l accu iacy or inacc i i rac \f such a model.

24. Research Project Use your school's l i b r a r y , ihe In t e rne t .or some other reference source io locale dam tha i you t h i n kdescribes a l inear re la t ionship. Create .; scal ier p in t of thedata and find the least squares regression line that repre-sents the po in t s . Interpret the slope and v - in te rcep i in thecontext of ihe data. Wri te a summaiA of \1 f i nd ings

Skills Review

In Exercises 25-28, evaluate the function at e;sch value ofthe independent variable and simplify.

25. /Ivl = 2.r- - .n - 5

i a ) / ( - 1 ) ( b ) / ' In1 + 21

26. t j t . v ) = M-- - n V + 1

i;n ,i,'!-2J ,{b) f>(- - 2!

In Exercises 29-34, solve the equation algebraical!}. Checkyour solution graphically.

29. 6.x - I - -9v - S 30. ? ( \- -i - 7.v - 2

31. S.v: - l (h - 3 = 0 32. lOr -- 2.\ 5 ~- 0

33. 2.v: - 7.i - 4 = 0 34. 2.r - SA - 5 - 0

82Chapter 1 Functions and Their Graphs

What Did You Lein?

Key Termsslope, p. 3

point-slope form, p. 5slope-intercept form, p. 7parallel lines, p. 9

perpendicular lines,p. 9function, p. 76domain,p. /6

range.p, 16

independent variable,p. 18dependent variable,p. 18function notation, p. ;sgraph of a function, p. 30Vertical LineTest,p.37even function,p. 36

Key Concepts

1.1 • Find and use the slopes of lines to write andgraph linear equations

1. The slope in of the nonvenical line through (.v,,.V|)and (A,, v,). where .v, # ,v,. is

__ v2 — v, ^ change in v

x2 - .v, change i n . v '

2. The point-slope form of the equation of the line lhaipasses through the point (x{, v , ) and has a slope of inis v - v, = »;(.v - .t,).

3. The graph of the equation v = nix + b is a linewhose slope is in and whose v-imercept is (0. h).

1.2 8 Evaluate functions and find their domainsJ. To evaluate a function/(.v). replace the independent

variable .v with a value and simplify the expression.

2. The domain of a function is the set of all realnumbers for which the funct ion is defined.

1.3 • Analyze graphs of functionsI, The graph of a function may have intervals over

which the graph increases, decreases, or is constant.

2. The points af which a function changes its increasing,decreasing, or constant behavior are the relative mini -mum and relative maximum values of the function.

3. An even funct ion is symmetric with respect to thev-axis. An odd function is symmetric with respect tothe origin.

1.4 B Identify and graph shifts, reflections, andnonrigid transformations of functions

J. Vertical and horizontal shifts of a graph aretransfonnation.s in which the graph is shifted up ordown, and left or right.

2. A reflection transformation is a mirror image of agraph in a line.

odd function, p. 36rigid transformation, p 47inverse function,p. 62one-to-one, p. 66

Horizontal Line Test, p. 66positive correlation, p. 74negative correlation, p. 74

3. A nonrigid transformation distorts the graph bystretching or shrinking the graph horizontally 01'vertically.

1.5 H Find arithmetic combinations andcompositions of functions

1. An arithmetic combination of functions is the SUBdifference, product, or quotient of two functions."domain oTthe arithmetic combination is the set creal numbers that are common to the two function

2. The composition of the function/with the functiug is

( /° tf)M =/U'M). I

The domain of / ^ # is the set of all x in the domaHlfof g such that g(,\) is in the domain of/

1.6 H Find inverse functions

1. If the point (a. /;) lies on the graph of/, then thepoint (b, a) must lie on the graph of its inversefunction/"1, and vice versa. This means that thegraph of/" ! is a reflection of the graph of/in theline v — -v.

2. Use the Horizontal Line Test to decide if/hasan inverse function. To find an inverse functionalgebraically, replace/(.v) by v, interchange theroles of .i and v and solve for v, and replace \v

/~'(.v) in the new equation.

1.711.7 Q Use scatter plots and find linear models. A scatter plot is a graphical representation of data

written as a set of ordered pairs.

2. The best-fitting linear mode] can be found using thelinear regression feature of a graphing util i ty or acomputer program.

JSv Exercises

i:j )„ Exercise- i iind 2. sketch the lines wi th the indicated30P&' thn'Uiih :!K' point on (he same set of the coordinate

In L\ert SLS 3-1. plot the two points and find the slope of[he line passing through the points.

in r\erds,.s ^-'8. f i i ) use tht1 point on the line and (he slopeof the line to find the general form oi' the equation of theline, and di) find three additional points through which theline pas^L- i Fhtrt are many correct answers.)

HI.

II .

12.

13.

14.

IS.

16.

17.

IS. :

In Kxeivises 19-22, find the .slope-intercept form of theiquiitinn nf the line tha t passes through the points. I'se aKraphiij" util i ty to graph the line.

Review Exercises

Kate of Change In Exercises 23-26, you are "i1-dollar value of a product in 2008 and the rate at v-1

value of the item is expected to change during *5 years. I'se this information to write a linear ujuauyives the dollar value V of the product in t trms o*/. (Let t = H represent 2008.)

83

t\

h ' t

27. Saic\! llic second ;uul ihird qiuirtL'r.s oi 'UxL' -L i in in ic rcc h U M n c s s luid s;ilo of Si 60.000 and!Vsjx 'CUvcl \ The ynnvlh ol sales tn l lmvs a ] J I Kf : s t ima ie sale-, durini : the 1'ourlh cjiiarlt.1

2S. Depreciation The dulhu ' i luc ol i p\ pi i\- $225. The prnduci w i l l dn.ii. ISL i n - U U L H

rate o!' S12 75 per sear.

' u i \Vri le a l i n c a i - e q u a l i u n tha i y i s e ^ [lie d o l l a r \ u ; ; ;I he i'JVD player in lerm.s of the \ear t i i ,e:represent 20(16.!

( h i L ' s e a graphins: u t i h t s lo L?raph Ihc equat icpai'i i a i . Be siire to ehoose in i p p m p n i t iu i n d o u . Slaie ihe d in iL 'ns ions nf mil I L U i n _and e x p l a i n \viiy sou eho^c iht il e i ' i

( c ! i_'se the \rJm- or innc f e a t u r e ui MHI _ i iphin c M ] m a t e [lie d o l l a r \e ol" the DVi ^i n 2010 Confirm y o n r a n s \ \ e r iLJ->i HL i l l

: d : ALX-urd iny In t l ie model , when \ \ i l l ihe DMhave no s a l Lie1.'

In Exercises 2M-32, write the slope-intercept form1-,equation.s of the lines through the given point (a) par;ilthe yiven line and (b) perpendicular tn the given line. \r result wi th a graphing uti l i ty (use a square st'tiia-


1.5 In Exercises l f l l - 110 . let/(.v) = 3 - iv.g(.v) = N i. andh(x) = 3x2 + 1. and lind the indicated values.

101. I / - , : ;)(-()

102. I /' ' / i M s i

103. ( ; -i elC.ii

104. ty inn >105. I /'//Id I

106. ( j ) ( l )

107. |/; . 1 - 1 1 7 ]

108. if! />! 21

109. I / / / ) ( -4)

110. It! /i 1 1 6 1

J In Kxercises 1 1 1 - 1 1 6 , lind t\vo functions / and £ sudi thatU A')(-v) — ll(x). (There are many correct answers.I

1 1 1 . Mil - (\ .* ) - '112. /;(.v] ^ (I - 2i r

1 1 3 . / i l l I = s 4! - 2

114. / i l . i l - s' ii 4 >'

4MS.

116. / /H I =13

s a n g t e A T ( v , ) a n dthe years 1990 through 2004 can he modeled by

y, = 0.00204/2 + 0.00 IS/ + 1.1121

and

v, = 0.02741 + I). 785

where / represents the year, w i t h / = 0 corresponding

1990. N . - l ! , , , - ', ' . M I L - , ' ! ! i l ' . . ' U i I \ : i : ~ : .:!:,•' :>•• ' ' . :\ I . i n , :

1.6 In Exercises 119-122, find the inverse function o f /informally. Verify thal/(/~' U-)) = .v and/''/(.r)) = .v.

119. / T . i ) - 6.1

120. / I l l --= i •• 5

121. . / ' l . v i = - ' 1 1 - : , :;

122. / u ) - I-7-4 ;!

In Kxcrcises 123 and 124, show that / and g are ift.nct.ons (a) graphically and (i,) numerically.

123. / ( . , ) ,:

124. /u-l --

Data Analysis In Kvercises 1 1 7 and U.S. the numbers (in 132. / ( t lmillions) of students taking the SAT ( i , ) and ACT ( v , ) for ^ ( ,

ic years 1990 thrnuoh 7II/IJ ' • • •

In Exercises- 125-128. use a graphing utility tofunction and use the Horizontal Line Test towhether the function is one-to-one and an inversexists.

125. / ' U l = - • i .i

126. /'in - [ i I I "

127. / id)/ - j

128. vlil : -. i""i-li

129. / l . i l

1311. / l i ) • -

131. , / d i

134. / ' ( e l -

1.7 In Exertre '" sets of da(J

a vc

, ^135.

y««

IF

u Cre u^ i i n i f

T; DHLS t i i L J in nslnp n o\n .v und i appeal- lo bejpprn\ im Ht.t\n u . \pl t in .

j,1g S/r«5 Tiit \e p i i as tested by benciny i i Atcniinic'iers 1 n t imes per m i n u t e u n t i l I L f a i l e L l( v equals the t ime to fa i lure in hours ) . The resul ts aregiven as the fol lowing ordered pairs.

(a) Create a scatter plot of the data.

(b.i Does ihe re la t ionship between A and v appear to beapproximately linear'.' If not . give some possibleexplanat ions .

139. Falling Object In an exper iment , students measured thespeed ,v (in meters per second; of a ball ! seconds af ter it\\as released. The resu l t s are shown in the tab le .

Time, t : Speed, 5

. " L \L- t he model from par: i c > tn esi imaic t he speed < > l:he ball af ler 2,5 seamjv

14(1. Sports The fo l l owing ordered pairs (.v, v) represent theOh. m pic year .v and the w inn ing s i me v ( i n m i n u t e s ) in I hemen's 400-meter freestyle s w i m m i n g event. v ' . , ; , • •

i l%4. 4.203) (19SO. :\S?^! 119%. 1S001

11968. 4.1501 I 1984. 3.854) (2000. ? . f i77 )

i 1972. 4.00?) ( 19SS. 3.7S3) (2004. 3 . 7 1 S )

i!9?6. 3.S66) (1992, 3 750)

Review Exercises 87

(a) Use the fi'i>re.\xiou leatun. 1a l inear model for the dat L\ \ i l h A — 4 correspond n 1correlat ion coefTicient fur tin

i b ) L ;se a g raph ing u t i l i t y to LI tdata.

Height In Exercises S41-144, the folltnU*,3') represent the percent v of women20 and 29 who are under a CLrtair "

rdere-1

en she ii t n

purs^ts oftcct).

Synthesis

True or False? In I^xercises 145-148, ciethe statement is true or false. Justify you;

145. i f the graph of the pareni t u n c u o n !''.->u n i t s to the r ight , moved three u n i t s i i - vin the A - a \ i s . then the p o i n t i - 1 , 2X; v .of the t ransformat ion .

146. I f /(.v) = A'" where / / is odd. / ' e x i s t - .

147. There exis ts no funct ion / s u c h t h a t /

148. The sign of the slope of a regre^i.

84 Ciiapter 1 Functions and Their Graphs

1.2 In Exercises 33 and 34, which sets of ordered pair;represent functions from A to fl? Explain.

33. A = {10. 2!)- 30. 40 \d fl - { ( ) . 2. 4. 6}

( ; i i {(20. 41. (40. ( I I . (20. 6) , (30, 2 ) \) { ( 1 0 , 4) . (20, 4) . (30. 4) . (40. 4) , '

(c) {(40. 01. (30, 2), ( 2 0 . 4 ) . ( 1 0 . 6) |

(d) {(20. 2). (10 . 0), 140.4)}

34. A = { i t . v, t f 1 and fl = I - 2. - 1 . 0. 1 . 2 j

( a j { ( r . - 1 ), (u. 2). ( i f . 0). (n, -1)\) { ( / / . - - 2 ) . I r , 2 ) , ( » • . l ) f

( O j i f f . 2 ) . ( r . 2i . ( t r , I I , ( i f . I ) ;

( i l l { ( » , -2). d . ( ) i . ( H - . 2 ) !

In Exercises 35-38. determine whether (he equat ionrepresents v as ;t function of'.r.

35. I6.r: « y- = 0

36. Iv - y - 3 - 0

37. v = s]~-7

38. |v| - A- -r 2

In Exercises 39—82, evaluate the funct ion at each specifiedvalue of the independent variable, and simplify.

39. /Ivl --= .\ f 1

( a ) / ( I )

(el /(V)

40. <;(.v) -•= v 4 '

( a ) . i f i H )

( c ) i , < ( - 2 7 )

45. /(.i) - „ 25 -";;.:

46. / ( A ) --- x i:~.'-~f6

49. Coat A hand tool manufac turer produces a pr,,,i,,,,i, ,u . . - • ' ' • •

( b ) Wr i t e the prof i t /J as a f u n c t i o n of A.

50. Consumerism The retai l sales R l i n b i l l i o n s of ilawn care products ;ind services in the Uni ted States1997 In 20(M can he approximated by the model ;

_ f O . I 2 f > / ; - O.S9/ + 6.H. 7 < t <

Ri!] ~ [ O . I 4 4 2 / ! - 5.61 }?- + 71 .10 ; -- 2S2.4. \ < I <

jf In Exercises 51 and 52, find the difference quotient isimplify your answer.

5 1 . / ( A ) = 2 i ;

52. /(A ) ,- A -

1.3 In Exercises 53-56, use a graphing utility to graphfunction and estimate its domain and range. Then finddomain and range algebraically,

53. /(.v) = ? 2.v:

54. /(.v) = , 2r"r"T

55. /; (.v) = ^ 36 ~"i:j

56. ,i , ' ( .v) •- \.\- 5 ^

In Exercises 57-60. (a) use a graphing ut i l i ty to graph theequation and (h ) use the Vertical Line Test to determinewhether y is a function of .v.

In Exercises 43-48. find the domain of tlie function.

57.

43. / ' (A) = - —4

f ;Bs fcrcrcisi. r.l_<,4 l , i l use a uraphin" ut i l i ty In iiraph the,Kl,anjn c i l 10 tkt.rmmc the ope,, intervals on which the

%MO»n.s» decreasing. ,,r constant.

•rases fo-ft<\e ;* ^raphmg uliliiy to approximate) decimal places) any relative minf inum or relativeurn values of the function.

In Exercises 69-72. sketch the «raph of the function hyhand.

72.

In Exercises 73-7S. determine algebraically whether thefunc t ion is even. odd. or neither. Verify vuur answer usir,» a§rapl]in!; u t i l i t y .

'••'' hi K\LTelse.s 79-84. identify the parent function anddescribe She transformation shown in She graph. Write miequat ion Tor the graphed function.

Kevicw exercises

In I'^xercises 85-SS, use the graph of v -,/{.v) t i:r;;pli the

In Exercises 89-Iii!!.related to one of the six parent functions on pa^e 42. i u )Identify the parent function/, (b) Describe the sequence oftransformations from / to h. (c) Sketch the ^ruph o f / / byhand, (d) Use function notation to write h in iern^ of theparent function f.

1.1 exercises - deer valley unified school district / …•/checkpoint now try exercise 57(b)....

Documents