# NEW WORK FOR SUBBU

strength
math_opt_op.pdf

17, r) -,r I : -l 19.t:...: 1 21.2t-31:4

/ 23. ^t: -rr - 1)t I,r r.= ll-vl 11 1

Evaluating a Function In Exercises 29-14, evaluate the function at each specified value of the independent variable and simplify.

29.fO-3t+t (a) f(2) (b) /(-4) (c) f(r + 2)

30. s(y) :1 - 3y (a) s(o) tul s(l) (c) s(s + 2)

tlzt.t(t):t2-2t (a) h(.2) ft) /,(1.s) (c) h(x + 2)

32. v(r) - !rr3 (a) v(3) (b) Y(;) k) v(2r)

33./(.-r)::-./y @) f(a) (u) l(0.2s) (c') [email protected])

3a.f(x)- aE+8+2 (a) f (-+) (b) /(8)

1

' x'-9 (a) q(-3) (b) q(z)

)t2+\ 36. ./(r)- t'

(il qQ) 0) q(o) lrl

37. i(r) : "'x ar f(e) (b) /(-e)

38. -.,. : -r *4 : , -i (b) /(_5)

-, - l. x<0 'lq -, - l. .r > 0'\-

'\- 6r t(0)

', - -:. -r < 0t. > |

1..

1OG Chapter I Functjons and Their Graphs

Testing for Functions Represented Algebraically In Exercises 77-28. determine whether the equation represents,r' as a function of r.

18.x:]2+1 20. y :-lf+ 5 22.r:-.y+5 24.r'l!2:3 20. lyl :1-x 28.r,:8

(c) /(x - 8)

(c) q(y + 3)

(.c) q(.- x)

(c) /(t)

(c) f(t)

(c) JQ)

,cl .f(l)

Evalr.rating a Fr.lnction In Exercises 45-48, assume that the domain of/is the setA = {-2, - 1, 0, 1, 2}. Determine the set of ordered pairs representing the function/.

[.rr-4. x<o42.fG)-1r_r,., r>0L1 - i.\ (il f?2) (b) /(0)

[.r- ]. t<0 I

a3.f(x)-1a. 0<r<2 L*, + t. r > 2

(a) .f(.-2) (b) /(1) (s - ), r < otJ

44. ffrr --]s. us r < I l.+*- r, r2 l

(a) J(. 2) (b) /(])

(c) /(1)

@ f(a)

(c) f(t)

a6. .f(.x) : x2 - 3 as. /(x) : lx + 1l

45. f(x) - x2 a7. f(x): lxl + z

Evaluating a Function In Exercises 49 and 50, complete the tahle.

4s. h(t): llr + :l

l" - ?l50..f(r) -:

Finding the lnputs That Have Outputs of Zers In Exercises 51-54, find all values of x such that/(r) = g'

st. 16) : 15 - 3x 52. f(x): 5r * I 3r-4

sa. f(x') - 2r-3s3. /(x) :

Finding the Dornain of a Function In Exercises 55-6J. find the domain of the function.

,l ss. fG): 5x2 + 2x - | s6. s(ir) : 7 - 2x2 4-3v

57. hhl - ' 58. ,s( r') -I y-) se. /(x) - 1C - 1 60. /(x) : X/" + 3x

. t 3 l0 { el. gtrt - ' - 62. h(r) - .., 1..I f t- i LA

r'*2 -,8+6 64./(:r) :--' o f .t

t -5 -4 -3 -1 It(r)

,t 0 l2 I 2

4

/(')

63. s(.v) : 5- 10

the Domain and Range of a Function In 1) mriffs 65-68, use a graphing utility to graph the hhu Find the domain and range of the function.

. ,.- -E' - \

+ i 66. f(x): 1F I 1 68. g(x) : I, - sl.j1- -r,-; : i1r + 3l

I. Geometry Write the areaA of a circle as a function of rs --ircumference C.

il" Cmmetry Write the arca A of an equilaterai tiangle "ts i tunction of the length s of its sides.

1!- E4loration An open box of maximum volume is to s made from a square piece of mateial, 24 centimeters cm a side, by cutting equal squares from the corners and uuia-e up the sides (see figure). ,"1 , The table shows the volume 7 (in cubic centimeters)

of the box for various heights x (in centimeters). L-se the table to estimate the maximum volume.

i Plot the points (x, I/) from the table in part (a). Does rtre relation defined by the ordered pairs represent V a-s a function of x?

; ff tr'is a function ofx. write the function and deterrnine ir-' domain.

; L -'.- a graphing utility to plot the points from the ::ile in part (a) with the function from part (c). FL.m'closely does the function represent the data? Erpiain.

Section 1.3 Functions 107

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

-I

Geometry A rectangle is bounded by the x-axis and the semicircle y - -66 - *'(see figure). Write the areaA of the rectangle as a function of -x, and determine the domain of the function.

74. Geometry A rectangular package to be sent by the U.S. Posta] Service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure).

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

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.

What dimensions will maximize the volume of the package? Explain.

73.

(a)

(b)

(c)

1

2

3

4

5

6

484

800

972

1024

980

864

x +24-2x- x

-6 4

krcreasing and Decreasing Functions In Exercises 5-26, determine the open intervals on which the hnction is increasing, decreasing, or constant.

Section 1.4 Graphs of Functions 1 1 9

ae. f(x): [, - 1n + 2 s0. /(x) : [, - 2l + \ sl. /(x) : [2,] s2. f(t) : [4xn Describing a Step Function In Exercises 53 and 54, use a graphing utility to graph the function. State the domain and range of the function. Describe the pattern of the graph.

53. s(x) :2Ex - ll+rll)tr+ x/ sa. s(x) : z(1* - [*,n)' Sketching a Piecewise-Defined Function In Exercises 55-62, sketch the graph of the piecewise-defined function by hand.

lI. -r-r + !2 : 25 6

x" .f(") :1*

5./(r):x3-3xz+2 4

4

!7. f(x) : 3 fi. f(*) : x2/3 3r. /(x) -- ,'G + Z

22. x2 :2xy - 1 4

2a. f(x) : x2 - 4x 3

-5

26. f(x) : lF - 1 7

{ ss. f(*):{?. * '' LJ-X. lx + 6.

56. l(*) : lz* _ +. l./tri

57. llx): I-;---' LV4 - x' [l - t* -58./{x) :l

-j LVx - L' lx + 3. I

se. f(x): j3. [2, - t. [x+5'

60. g(x) : l_2. [s* - +, fz* + t.6t..flx):1, lx'- /' (^

62. h\x\: ]', I :' L"{- T r.

63. f(x) : s 6s. f(,*) : 3x - 2

,f el. n(x) : x2 - 4 6e. f(x) : Jl -x 7t. f(x) : lx + 2l

x<0 x>0 x<-4 x>-4 x<0 x20

i)', x<2 x>2

x<0 0<x<2 x>2

ir : -J -3 < x < 1

x> 7 x<-7 x> 1 x<0 x>0

-4 ffi -l

Increasing and Decreasing Functions In Exercises v7-i4, (a) use a graphing utility to graph the function ud (b) determine the open intervals on which the function is increasing, decreasing, or constant.

28. f(x) - x 30. /(x) -f/q 32. f(x) : -I - ,

33. "f(r) : l, + 1l + lx - 1l

-ta."f(r) : - lx + 4l - lx + 1l

Approximating Relative Minima and Maxima In Frercises 35-46, use a graphing utility to graph the function and to approximate any relative minimum or relative maximum values of the function.

lx.f(*): x2 - 6x 36. f(x):3xz -2x- 5 ln. l,:2x3 + 3x2 - 12x 38. y: x3 - 6x2 + 15

39. lz(x) : (* - I)-/i a0. s(x) : [email protected] - * AL f(x) : x2 - 4x - 5 a2. f(x) : 3xz - 12x {r. /(x) : x3 - 3x a. f (x) : - x3 + 3x2 {5."f(r):3x2 -6x* I a6. f(x):8x- 4x2

t-. l-i[6ry of Parent Functions In Exercises 47-52, dietch the graph of the function by hand. Then use a graphing utility to verify the graph.

a7. f(r) : fixn + z a8. f(i: [,n - 3

Even and Odd Functions In Exercises 63-72, ase a graphing utility to graph the function and determine whether it is even, odd, or neither.

Think About lt In Exercises 73-78, find the coordinates of a second point on the graph of a function/if the given point is on the graph and the function is (a) even and (b) odd.

tt. (-), +) 7s. (4,e)

77. (x, -y)

6a. f(x) : -e 66.f(x):5-3x 68. /(-r) : -x2 - 8 70. sQ):11 * | 72.f(x):-lr-51

t+. (-j, -t) 76. (s, - 1) 78. (2a,2c)

-4

120

79.f(t):t2+2t-3 ./sr. g1r1 : x3 - 5x

83. /(x) : x{-P 85. s(s) : 4s2/3

87.f(x):4-x 89. /(x) : x2 - 9

93. l , It-L

Chapter I Functions and Their Graphs

Algebraic-Graphical-Numerical In Exercises 79_g6, determine whether the function is even, odd, or neither (a) algebraicallS (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare /(x) and /(-x) for several values of r.

Finding the lntervals Where a Function is positive In Exercises 87-90, graph the function and determine the interval(s) (if any) on the real axis for which f(*) > 0. Use a graphing utility to verify your results.

80. /(x) : .{6 - 2x2 + 3 82. h(x): 13 - 5 8a. fQ) : t,G + s 86. /(s) : !c3/2

88. /(x) : 4r t 2 90. .f(*) : x2 - 4x

an overnight package from New york to Atlanta is \$18.80 for a package weighing up to but not including I pound and g3.50 for each additional pound or portion of a pound. Use the greatest integer function to create a model for the cost C of overnight delivery of a package weighing r pounds, where x > 0. Sketch the graph of the function.

94. ) 4

./ss. fuonELtNG DATA The number N (in thousands) of existing condominiums and cooperative homes sold each year from 2000 through 2008 in the United States is approximated by the model

N : 0.482511 - t1.293t3 + 65.26t2 - 48.9/ + 578, 0<t<8 where / represents the yeal with t : 0 corresponding to 2000. (Sonrce: Narionai Association ol Realtors)

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

(b) Use the graph from part (a) to determine during which years the number of cooperative homes and condos was increasing. During which years was the number decreasing?

(c) Approximate the maximum number of cooperative homes and condos sold from 2000 through 200g.

96. Mechanical Engineering The intake pipe of a lO0-gallon tank has a flow rate of 10 gallons per minute, and two drain pipes have a flow rate of 5 gallons per minute each. The graph shows the volume V of fluid in the tank as a function of time /. Determine in which pipes the fluid is flowing in specific subintervals of the one-hour interval of time shown on the graph. (There are many correct answers.)

10 20 30 40 50 60 Time (in minutes)

E

e

g

91. Business The cost ofusing a telephone calling card is \$1.05 for the first minute and \$0.08 fbr each additional minute or portion of a minute. (a) A customer needs a model for the cost C of using

the calling card for a call lasting / minutes. Which of the following is the appropriate model?

Cr(t):1'0s+0.08[/-1n

czo : 1.0s - 0.08[-(/ - l)n (b) Use a graphing urility ro graph the appropriare

model. Estimate the cost of a call lasting lg minutes and 45 seconds.

E SZ. Whyyou shoufd t's*rn it (p. t tTl The cost of sending

=

:

J Using the Graph of a Function In Exercises 93 and.94, write the height ft of the rectangle as a function of x. r00

a0

e50 (.)

'.i ,(

t, -i f +r - I

{60, 100)

(45, s0)

(30. 25)

Chapter I Functions and Their Graphs

Vocabulory and Concept Check

1. Name three types of rigid transformations.

2. Match the rigid transfbrmation of -r, : /(x) with the correct representation, where c > 0.

while a reflection in the -v-axis of y : /(;r) is represented Ay h(,x) - 4. A nonrigid transfbrmation of y : /(x) represented by cl(x) is a vertical stretch

when

-

and a vertical shrink when

Procedures and Problem Solving

(a) /z(,r) :./(x) + c (b) h(x) : JQ) - , (c) h(:r) - .f (.' - c) (d) /z(r) : /(.t + c)

In Exercises 3 and 4, fill in the blanks.

Sketel':ing Tr*r+s\$srm*tions In Exercises 5-18, sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utitity.

5. r(x) : "{ s(r) :x-+ h(x) : 3,

7. f(r): 12 g(x) :x2+2 h(r')-6-2)'

e. fQ) - x2 s(x) : -*+I h(.r) - -(x - 2)z

tt. 1Q) : "z

s(r) : jr' h(r) : (2x)2

13. l(x) : lxl g(;r):l"rl -t /z(x) : lx 3l

15. /(r) : '6 g(r) : -,G +l h(r):.,8-z+t

I t7. f(.r) : -

,ot:1 *z

h(x\: | * 2' x- |

(i) horizontal shift c units to the left (ii) vertical shift c units upward (iii) horizontal shift c units to the right (iv) vertical shift c units downward

e. [email protected]: L* s(r) :)x+2 h(*)-ia-zl

8. /(x) : ,z g(r):x2-4 h(*)-(x+2)2+1

10. /(-r) : (x 2)' s(x) -(x+2)2+2 n(i: -(-r - 2)'- t

12. f(x) : x2 o("1:1r2+2+" n(i - i*'

la. /(x) : lxl s(r) -lx+31 h(*):-2lx+21 -1

16' JQ) - Jr s(i : i'G h(x):-Jx+1

I 18. l(:r) : -

x

I

s(r):- 4 .r

Sk*t*hing Tr*nsf*rnrnti*::s In Exercises 19 and 20, use the graph oflto sketch each graph. To print an enlarged copy of the graph, go to the website www.mcthgraphs.com.

19. (a) y-.('lx)+2 (b) v : -.f(,) (c)v-l(r-z) (d) r:/(x + 3) (e) y : 2/(r) (0 r - l(-x) tel ., : /(i,)

20. (a) I :./(r) - 1 (b).y:l(x + 1) (c)r:"r(,-l) (d)y:-f(.r-2) (e) r:l(-r') (f y : jl("r) tor .,: fl 1r')

J -','

(1,2)

r-l i-r

(-2,4)

f

Err*r Analysis In Exercises 21 and, 22, describe the error in graphing the function.

21. 1(r): (r + 1)2 22. f(x) : (, - 1)' ]

.l/rz. o r\z:+

(1,0) (0. - 1)

+-i"'---"-i I

r,L

3. A reflection in the r-axis of y : .f(x) is represented by h(x)

x*3/,(r)

Section 1.5 Shifting, Reflecting, and Stretching Graphs 129

lx:*:.G+z !5. .r : (* - 4)' ,F ----1-.)ir, |

- A L

F.' g5r"r, of Parent Functions In Exercises 23-28, {+5. onFare the graph of the function with the graph of its prent function.

24. y:! - sx 26. y: lx + sl 28. y : -,G=a

f= tibrary of Parent Functions In Exercises 29-34, *[email protected] the parent function and describe the transformation *orn in the graph. Write an equation for the graphed fuetion.

D.530.4

-4

ltr. 5

1

ltt.

Rigid and Nenrigid Transforrnations In Exercises *846, compare the graph of the function with the graph dits parent function.

/35.r,:-lrl rl" -r: (-r)2

:p.r:a t.a. n1x) : +lxl

{3. g(x) : a1x3 as. "r(r)

: -,4; Rigid and Nonrigid Transfcn*aticns In Exercises 47-50, use a graphing utility to graph the three functions in the c me yiewing window. Describe the graphs of g and h rdative to the graph of/.

t7" f(*) : x3 - 3x2 48. .f(x) : x3 - 3x2 + 2 s(.r) : f(x + 2) s(x) : f(* - r) h(i : +i\t h(x) : f(3x)

!i- 20'10/used ufder llcense fr0m Shu11-"rst0ck enm

Deseribing Transfarmations In Exercises 51-64, g is related to one of the six parent functions on page 122. (.a) Identify the parent function/. (b) Describe the sequence of transformations from;f to g. (c) Sketch the graph of g by hand. (d) Use function notation to write g in terms of the parent function/.

s1. s(x) :2 - (x + 5)2 s2. [email protected]): -(, + 10)2 + s 53. s(x):3+2(x-4)2 sa. g(x): -iA+T2 -2 ss. s(x) : 3(x - 2)3 s6. s(x) : -j(x + t)3 s7.sG):@-t)3+2 s8. s(ir) : -(. + 3)3 - 10

f(.):x3-3x2 s(x) : -\tt,) h(*) : f(-*)

50. /(r) : x3 - 3x2 + 2 s6) : -fG) h(x) : f(2x)

60.s(x)- f -nox- I 62.sG):+1,-21-3 6a.g(x):--/xtt-e

-8

32.

34.

1

se. s(r) : '' - e,Y-t6 61. s(x) : -21* - 1l - 4 63.sG):-i-/.+z-r

65. M*SELING DATA The amounts of fuel F (in billions of gallons) used by motor vehicles from 1991 through 2007 are given by the ordered pairs of the form (t, f(t)), where r : 1 represents l99l . A model for the data is

F(t) : -0.099(t * 24.t12 + 183.4. (Source: U.S. Federal Highway Administration)

(1, 128.6)

(2,132.9)

(3, 137.3)

(4, 140.8)

(s, r43.8) (6, t47.4)

(1,1s0.4)

(8, 1ss.4)

(e, t6t .4) (10, 162.5) 0q,fiZ.S) (11, 163.5) (1s, t14.8) (12.168.1) (16, 175.0) (i3, 170.0) (17, ti6.t) (a) Describe the transformation of the parent function

[email protected]: P. (b) Use a graphing utility to graph the model and the

data in the same viewing window. (c) Rewrite the function so that / : 0 represents 2000.

Explain how you got your answer.

36. y - l-xl 38. v : -r: 40.v:-1

x

42. p(x) : \x2 aa. y -- 2.G +e. y: li{

A,r/ t 2

2 3

-l

Section .l .6 Combinations of Functions 137

: : sulary dffid Ceffie€pf'CC*ee\$*

.:-i:es l-4, fill in the blank(s).

,.lctions./and g can be conrbined by the arithrnetic operations of and to create new functions.

of the function.l with the firnction g is (./. .cxr) : .l(C(-r)) : - :1i3rr1 of .1 " .C is the set of all .r in the domain of g such that _ is in . r.iin of.f.

:- ,r]lpoSe a composite function. look lbr an _ and an function.

: - . . S)(r) : 71r: + 1). what is g(;r)'/

: : = Cures and Pro&Jerm \$o/wfngr

-; r-: :Lj:il *flTrav+ f*:r:r-ti*r:* In Exercises 7-10, frv"tl:-ratiitq +r:;+.rici:r**ire {*ml;!*+qj+* *f Sr,i*:sti*ft* In ': rraphsof/andgtographh(x)= (/+g)(-r).To Exercises 19-32, evaluate the indicated function for

' , enlargedcopyof the graph,gotothe website f(i=f -l andg(r) =x -2algebraicalll,. Ifpossible, - -:itqraphs.com, use a graphing utility to verify your answer.

-l- l -* -t 23

.: :, -:i*=cti< '[*r::hi::;r'*i*..r: c]f Furrcti*a:: In

.=. 11-18, find (a) (.r+c)6), (b) (/-s)G), , . and (a) (/SXr). What is the domain ot flg? :'-3, g(r) :.t-3 -lr-5. s(x):1 r :,r. g(r)-1-x

\ .)l1l: \A \-./

5. g{-i)-.,/t -,

-rrr 4. s(r) :*

-1

* re ph i r:q i+ * & rit* ;-* elic il*r-s: i=i r:*ti*n *f f qJ i-.slii-=:: : Exercises 33-36, use a graphing utility to graph functions.f, g, and /z in the same viewing windou..

19, (.f + s)(3) 2t. \f s)(0) zt. (fg)(6) 2s. (fls)es') 27. (f dQt) zs. (fg)( 51) tt. (f'lg')(-t)

20. (f .s)( 2) 22. (..f + 8x1) za. (.fg)( 4) 26. (fls)\$) 28. (.f + d(.t 4) 30. (/s)(3/) 32. (f ls)k + 2')

In the

3 33. l(:r) 34' f(') 3s. /(-r) 36. .f(r)

38. /(-r) :

3e. /(x) : a0. f(r) -

*r**hir:+ * lur: +i l--li:*=: In Exercises 37-40, use a graphing utility to graph /, g, and f + g in the same viewing windorv. \l:hich function contributes most to the magnitude of the sum when 0 < x < 2? Which function contributes most to the magnitude of the sum when x>6?

37. ./'(i) - 3r, s(, - - l0

: lr. g(r) :,r - 1. ft(.r) : f(r) + g(-t) : 1-r, s(-t) : -r * 4. /z(r) : f(r) ,c(x)

g(.r) - l.r. /ir.rl-/lr) 'g{.r) = 4 -.r2. g(rr - r. /rrr) :/rr)/g(,\)

:tC

- l.I(-r): J-r+-:

e(-r) - -3r'2 - 1

I g(-r )

rl I---

I

) L-

g(r) - :r3

Chapter 'l Functions and Their Graphs

Vocobulary ond Conceqt Check

In Exercises 1-4, frll in the blank(s)'

1. If/and g are functions such that/(80)) : x and g(/(x)) : r' then the function g is the function ofl, and is denoted bY _-- '

2. Thedomain of/is the of-f 1, and the of/-t is the range of f, 3. The graphs of f afif 1 are reflections of each other in the line 4. Tohaveaninversefunction,afunction/mustbe-;thatis'/(a)

:[email protected] implies a: b'

5. How many times can a horizontal line intersect the graph of a function that is one-to-one?

6. Can (1, 4) and (2, +1 Ue two ordered pairs of a one-to-one function?

Procedures ond Problem Solving

Finding lnverse Functions lnformally In Exercises 7-14, find the inverse function of / informally' Yerify that .f (.f '(x)) = x and/-l( f (*)) = x.

Verifying lnverse Functions Atgebraically In Exercises tg-iq, show that / and g are inverse functions algebraically. Use a graphing utility to graph/ and g in

th-e same viewing window. Describe the relationship between the graPhs.,f t. f(x) : ex

{ s.f(r):x*7 tt. f(x) : 2x + 1 13. f(r) : i,6

8. /(r) : 1x 10. l(x):x-3 12. f(x) : (x - \)la t4. f(x): xs

(b)

ldentifying Graphs of lnverse Functions In Exercises 15-18, match the graph of the function with the graph of

its inverse function. [The graphs of the inverse functions

are labeled (a), (b), (c), and (d).1

,l ts. f(*) : *', eQ) -- 1G 20. f(.) _- :, g(x) :x 21. f(t) : J* - a; s(x) : x2 + 4, x > o 22. ltrl -- 9 - 12. x > 0: g{x) : ./q - x

/ zl. tQ) : t .r'. s(r) -- { - r x>o; s(x):? o<x<1

Algebraic-Graphical-Numerical In Exercises 25-34, show that/andg are inverse functions (a) algebraically, (b) graphically, and (c) numerically.

2s. f(r) : -f,, - z, s(ir) : - 2x*6

r-9 26.flxt:

^ g(x):4r-9

27. 1(xl-xj-l 5. g(x) :f r-5

zs. tQ) : i, sG) : 16 2s. fG) : -rT-, s(r) : 8 t12, x < o

30. ./(x) : 1trx - 16, s(x) :

=

Q

3t. f(r) : 2x, g(x) = !2 32..f(x):r - 5, s(r) : x + :

, x-l 5x-l ,/ ll. 1t*l:

", , R(r) : - r - l

r-l-l 2xl3 3{./(x) :*_, P(r)-- -- I

2a. f(r)

(a)

(d)(c)

t6.15.

7

-1

4

-4

4

-4

7

_,1

7

1

7

-1

4

I

4

4

-4

17. 18.

trar*iffingWhether Functions Have lnverses In Exercises .G--lE. does the function haye an inyerse? Explain.

#- Dotnain Range 36. Domain Range

Section 1.7 Inverse Functions 149

Analyzing a Piecewise-Defined Function In Exercises 57 and 58, sketch the graph of the piecewise-defined function by hand and use the graph to determine whether an inyerse function exists.

lx'. o <.r < I 51. tlxl: <

[x. .r> 1

58. /(x) : [t- - ']t 'r < 3

l(x - 4)i. ;t' 2 3

Testing for One-to-One Functions In Exercises 59-70, determine algebraically whether the function is one-to- one. Verify your answer graphically. If the function is one-to-one, find its inverse.

s9. f(x) : x4 60. s(-r) : x2 - x4

3x*4 6t. flr): _) 62. f(i: 3x * 5

1

63. f6): x' 4

64. h\x): .x' 6s. .f(x): (a + 3)2, 66. q(x): (x s)',

,f ez. f(*) : J2" + 3 6s. /(x) : -C - 2 6e.fk)-lx 2 . x<2 70. f (x) : x2+1

Finding an lnverse Function Algebraically In Exercises 71-80, find the inverse function of/ algebraically. Use a graphing utility to graph troth f and f-r in the same viewing window. Describe the relationship between the graphs.

,ftt.f(x):2x-3 72. f(x) -- z,

/ ts. fG) : *' 7a.f(x):x3+1 75. f(x) : x3/s 76. f(x) : x2, ,x > 0

,/tt. Itxl: J4-=. o < x < 2 78. ltx) : J16 -?. -4 -< x < 0

4 7e. f(x) : -

x

80. f(x) : 4 Jx

I can --------- \$1 6 cans -------- \$5

il cans

-\$9 l-l cans ---..--- \$16

ll2hour _>\$40 lhow/-\$70

2 hours -' -\$720 4hottrs/

40.#.

42.

.tr. r-3, 6), (- 1, s), (0, 6)] JL - rr. 4), (3,1), (1,2)j

fucognizing One-to-One Functions In Exercises 39-44, &ermine whether the graph is that of a function. If so, &termine whether the function is one-to-one.

L Using the Horizontal LineTest In Exercises 45-56, use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function b one-to-one and thus has an inverse function.

x>-3 x<5

a

{5./(.r) :3-i, x2

x']_7

a6. f(x): lG * 2)' - t r', -.

48. g(x) :- - .^o,r" te. h(x): \n6 -? so. /(x) : -2xJTe -7 sl. /(r) : 10 s2. f(x): -0.65 53. s(x) : (, + 5)3 54. flx) : xs - J s5. /z(x) : lx + 4l - l, - 4l

l* 6l lx -r ol

Chapter 2 Solving Equations and lnequalities

Vocabulary ond ConcePt Check

In Exercises 1-4, fill in the blank.

1. A(n) is a statement that two algebraic expressions are equal' 2. A iinear equation in one variable is an equation that can be written in the

standard form

3. When solving an equation, it is possible to introduce a(n) --

solution,

which is a value that does not satisfy the original equation'

4. Many real-life problems can be solyed using ready-made equations called

5. Is the equation x 'l 1 : 3 an identity, a conditional equation, or a contradiction?

6. How can you clear the equation 1* , : j of fractions?

Procedures and Problem Solving

Checking Solutions of an Equation In Exercises 7-10, Solving an Equation lnvolving Fractions In Exercises determine whether each value of x is a solution of the 17-20, solve the equation using two methods. Then

explain which method is easier.

Values

1(a)r--; ib,1x:4I 1

(c)x:0 (d):r:;

r61198.r*;:i @)r:-2 (b)x:1 1

(c) x: i (d) x: 7

t.+1 + 3:4

fr'*': 810' .l :

equation.

Equation

,, 1,!:. " 2x -rr

'r.# -+:4 tt. ;t - ro:6 ,r.+*t,:1 ,r.+-2r:f

(a) x: -3 (b) .t: 0

(c) r:21 (d) x: 32

(a)-lr:-16 (b)x:a

(c)r:9 (d) x:16

Solving Equations In Exercises 2l-10, solve the equation (if possible).

21.3r-5:2x+1 22. 5r-3:6-2x 23.3(y - 5) :3 * 5r' 2-1. 5(:-4) + 4z:5 -62

txr^5r1 1,/zs.:r-;- 3 20. 1- r-r-, 2lz-41 - 3.r I1r :--r- - i - l0: 28. ^ -it, - 2) - l0Lt' -5 '' r + 100 - 4rr 5ir - 629.- - -6

-t+

17*r' 3l-r 30.

-

:100 11'

Classifying Equations In Exercises L1-16, determine whether the equation is an identity, a conditional equation, or a contradiction.

ll.2(x - 1): 2r - 2 12. x2 - 8x + 5 : (r - ,l)2 - 11 13. -5(x - 1) : -5(.r- + 1) 14. (x + 3)(x 5) : -t' - 2(x + 7)

14x 15.-1-i- -x-l x- l

5l 16. :+-:21xx

5.r--l l 31.--:l

1110 1a

-

- -

""'.-i .t-3 12-9 134

J+. ^T-.\'-l r r-J x2',r-6 12

35.-+ -:0r r-)

lOx*3 I a1

-:-

5r*6 2

2 36.3-2-t

-1.tL 12

L-

x-4 r-237. 6-a)(x-2)

Chapter 2 Solving Equations and lnequalities

Vocabulory and Concept Check

In Exercises I and 2, fill in the blank(s). 1. The points (a,0) and (0, b) are called the

-

and

respectively, of the graph of an equation.

2. A

-

of a function is a number a such that f (a) =

In Exercises 3-6, use the figure to answer the questions.

3. What are the x-intercepts of the graph of y : /(x)? 4. What is the y-intercept of the graph of y : S(x)? 5. What are the zero(s) of the function/?

6. What are the solutions of the equation/(r) : S(x)?

Procedures ond Problem Solving

Finding x- and y-lntercepts In Exercises 7-16, find the x- and y-intercepts of the graph of the equation, if possible.

Figure for Exercises 3-6

Verifying Zeros of Functions In Exercises 2l-26, the zero(s) of the function are given. Verify the zero(s) both algebraically and graphicallY.

0.

{ z.y:.t-5 2

-2

8.y:-1,-: 1

7

10.v:4-x2 5

,1

12. y: -\*-/, + Z 4

18.),:4(x+3)-2 20. y: l0 + 2(x - 2)

+1

-5

2

4 3x-l13.y:- ll.Y: ,-x+x 15. xy -2y- x* 1:0 16. x)-jr+4Y:0

Appro,xirnating x- and y{nlercepts In Exercises l7-20, use a graphing utility to graph the equation and approximate any .r- and y-intercepts. Verify your results algebraically.

17. y:3(x - 2) - 6 19. y: 20 - (3x - 10)

Function

zt. f(x) : 4(3 - x) 22. fG): 3(x - s) + e

{ zl. f(*) : x3 - 6x2 + 5x Za. f(r) : x3 - 9-r2 + 18x

r-r2 r- I ,5 /'(r'):--

-- IzJ.J\^'

3 5

26.f(x):r-:-Q x

Zero(s )

x:3 --') x:0,5,1 -lr : 0, 3,6

x:1

.: -) 5

Finding Solutions of an Equation Algebraically In Exercises 27-40, solve the equation algebraically. Then write the equation in the form /(x) = 0 and use a graphing utility to verify the algebraic solution.

27. 2.1x - 0.4x: 1.2 28.3.5x - 8:0.5;r 29. 12(x + 2) : 15(r - 4) - | 30. 1200 : 300 + 2(:r - 500)

3xl-2xl,r.; +r(x-2) : lo 32.; r1k- 5):6 33. 0.60x + 0.40(100 - x) : 1.2 34. 0.75x + 0.2(80 - x) : 20

x-3 3x-5 - x- 3 x-53s.-- , 30. x - n r-5 .r -^ x- 5 .l-3 ,37.i -;-,0 38. ro - s :r

39. (x+2)':i-6x*1 40. ("r + 1)2 + 2(x - 2) : (x + 1)(x - 2)

y:x2+2x-l 2 5

11.v

1

:x-E+z 4

-5

-6

Section 2.5 Solving Other Types of Equations Algebraically 207

/ocabulory ond Concept Check

L: Erercises I and 2, fill in the blank. L The general form of a

-equation

is

,1,,.\" 1 0, ttr'' *'' + arxl + arxl ttn: 0.

l. To clear rhe equarion 1 * , :

;| "t fractions. rnuitiply each side of the equation by

the least common denominator

-1. Describe the step needed to remove the radical from the equation -E + 2 : *. .t. Is the equation ra - 2x + 4 : 0 of quadratic type?

)rocedures and Problem Solving j: ving a Polynomial Equation by Factoring In :rercises 5-10, find all solutions of the equation i-:ebraically. Use a graphing utility to verify the ,,lutions graphically.

-i. J.ra - 1612 :0 6. 8xa - 18x2 : 0 -. 5t-3 + 30x2 + 45x : 0 \. 9.ta - 24x3 + 16"t2 : 0 q. t'*5:fx'+x

'rr. rf _ 2x3 : 16 * gx - ,lx3

33.2x+9,G 5:o 34. 6x-7tG-3:0 35. v[- .,6-s : r 36.,4+-,6-20-10 37. 3"'f - 5 - "( - t - o 38. a.[r - .t - "Gi - tz - -t

Solving an Equatian lnvolving Rational Exponents In Exercises 39-48, find all solutions of the equation algebraically. Check your solutions.

i:rving an Equetion of Quadratic Type In Exercises 43. (-r 9)z/t - r, 44. (r - 7)z/: -, l1-1{, find all solutions of the equation algebraically. +S. (r, - 5x - 21trt -2 46. (r2 , - 22\+/t : 16 -heck your solutions'

47. 3x(x - r)ttz + 2(r - r):/u - o Itr.,rr- 4x2 +3-o 12. ra - 5x2- 36:0 4g. i'.2(x_ 1)rl: +6:r(_1 1).rl: -o ,-1. -l,ra - 6512+ 16 : 0 14. 36t4 + 29t2 - 7 : 0

20.3-/i 6:0 22..Oxj.+3:0 24. ttxj t - 5:0 26.!4r-3+2:o 28.x-.r4i-:t:s 30.,4+5*2x:3 32..,G-5:r7--5

39. 3xt/3 i 2x2/3 - 5 { +t. (, - 5)zrz - ,u

solutions.

1t/ss..--*; TI

1l)\\ -----')r x*1 20-x

51.

-

- "\x

40. 9P/3 -t 24tt/t : - 16 42. (x - 1):r: - ,

45-r 54. -::;xJO

1 ir

i "a phical Analysis In Exercises 15-18, (a) use a graphing .:rilitl' to graph the equation, (b) use the graph to ioproximate any r-intercepts of the graph, (c) sety = 0 ind solve the resulting equation, and (d) compare the :rult of part (c) with the x-intercepts of the graph.

l,<, r : 13 - 2x2 - 3x 16. l: 2x4 - l5x3 i 1812 i-. ,r - r1 - 10.t2 + 9 18. v::ra 29x2 + 100

!: ving an Equation lnvolvlng Radieals In Exercises -q-38, flrnd all solutions of the equation algetrraically. ,-.heck your solutions.

i.r. -1,6 - 10 : 0 :1.--10-4:O 1-r. q2-t+5+3:0 :-r.i2-r+1+8:0 :-.-5-t-26-t4:x :,.r. -t+1-3x:1 -:1. ..r- l-.r5,, ll

Graphical Analysis In Exercises 49-52, (a) use a graphing utility to graph the equation, (b) use the graph to approximate any r-intercepts of the graph, (c) set y = 0 and solve the resulting equation, and (d) compare the result of part (c) with the r-intercepts of the graph.

49.t:-,/llr-30-, 50. v: 2r _ ./15 _ +, 51. r: "r1ar6

- . tt - lO - 2 52. .y ::;r - 3-rG - l

Solving an Equation lnvolving Fractions In Exercises 53-64. find all solutions of the equation. Check your

43 56.

-- j:t

r-l l .Yf t

58.4rt1:

Settton 2.4

- : n.; :** *ci*r*r::tit F*rrr-tr:f* In Exercises 53-60, use :re Quadratic Formula to solve the equation. Use a .raphing uti\ity to verity your so\utions graphica\Iy.

",i-1. I i 2.r -r2 : 0 54. .r2 - 10x i 22 : O , -r5, -,19-12 + 28x 1:0 56. 9,r2 - lS_t F 9 - 0

5-, r.: + 3x : -8 58. -r2 + 16 : -5r -.q. Ji.r * 16-r'f 11 : 0 60. 9f - 6x i 37 - o

: - '.,;::= e *uc*{r*i!; iql:+tir:* In Exercises 61-68, solve .:r equation using any convenient method. -1, r-r - 2,r I : 0 62. lt;r2 - 33x: 0 -1. I,r + 3)2 : 81 64. (-r - 1)r : 1 -5. ir 2t + f :0 66.,rr + 3-r j: O --. (.r * l): - .': 68. rrr,r2 bz - 0.. * 0

: .-=i-g x-irt*rcrg:ts &lg*hr*i<*liy In Exercises 69-72, :nd algebraically the x-intercept(s), if any, of the graph t the equation.

2

-1.

.quations …