1OG
I
Chapter
Functjons and Their Graphs
x<o
r>0
for Functions Represented Algebraically In
Exercises 77-28. determine whether the equation
represents,r' as a function of r.
[.rr-4.
42.fG)-1r_r,.,
i.\
17, r) -,r I :
[.r- ]. t<0
0<r<2
a3.f(x)-1a.
L*, + t. r > 2
(b) /(1)
(a) .f(.-2)
(s - ), r < o
us r <
44. ffrr --]s.
Testing
,r
1
1
)t
ll-vl
r.=
11
18.x:]2+1
20. y :-lf+ 5
22.r:-.y+5
24.r'l!2:3
20. lyl :1-x
-l
19.t:...:
21.2t-31:4
/ 23. ^t: -rr -
L1 -
(il
I
(c) /(1)
(b) /(0)
I
I
l.+*-
Function In Exercises 29-14, evaluate the
function at each specified value of the independent
Evaluating a
(a)
variable and simplify.
J(.
r, r2
2)
l
(c) f(t)
(b) /(])
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/.
29.fO-3t+t
(c) f(r + 2)
(a) f(2)
(b) /(-4)
30. s(y) :1 - 3y
(c) s(s + 2)
(a) s(o)
tul s(l)
tlzt.t(t):t2-2t
(a) h(.2) ft) /,(1.s) (c) h(x + 2)
32. v(r) - !rr3
(b) Y(;)
(a) v(3)
k) v(2r)
33./(.-r)::-./y
(u) l(0.2s) (c') J@f)
@) f(a)
3a.f(x)- aE+8+2
(c) /(x - 8)
(b) /(8)
(a) f (-+)
-
45.
f(x)
a7.
f(x):
x2
a6. .f(.x)
lxl + z
as. /(x)
4s.
36. ./(r)
-
(c) q(y +
50..f(r)
0)
-4
-5
-:
l" -
(b)
-, - l.
+
1l
50, complete
?l
I2
l
2
4
: 15 - 3x
3r-4
s3. /(x) :
st. 16)
(c)
-r
<
>
52.
f(x):
sa.
f(x')
-
*I
2r-3
5r
f(t)
Finding the Dornain of a Function In Exercises 55-6J.
find the domain of the function.
x<0
.r >
6r t(0)
1..
Finding the lnputs That Have Outputs of Zers In
Exercises 51-54, find all values of x such that/(r) = g'
(c) /(t)
/(_5)
-, - l.
', - -:.
t.
lx
-1
-3
0
,t
(.c) q(.- x)
q(o)
f(e) (b) /(-e)
38. -.,. : -r *4
'\-'\-
3
/(')
ar
'lq
3)
t'
: , -i
-
It(r)
)t2+\
(il qQ)
lrl
37. 'x
i(r) : "
:
x2
h(t): llr + :l
t
1
(b) q(z)
:
Function In Exercises 49 and
Evaluating a
the tahle.
'
x'-9
(a) q(-3)
@ f(a)
tJ
28.r,:8
1
f?2)
0
(c) JQ)
,l
0
|
,cl .f(l)
fG): 5x2 + 2x - |
4-3v
'
57. hhl I
ss.
se.
{
. el.
/(x)
-
1C
gtrt
-
I
63. s(.v)
t'
-1
f
3
-t-
: r'*2
5- 10
s6. s(ir)
:
7
-
2x2
- y-)
60. /(x) : X/" + 3x
l0
62. h(r) - .., 1..
i
58.
,s(
r')
LA
64./(:r)
'
:---,8+6
f
o
.t
Section
the Domain and Range of a Function In
65-68, use a graphing utility to graph the
mriffs
hhu
Find the domain and range of the function.
.
-E' -
.j1- -r,-; :
I.
\
,.+
66.
i
i1r +
f(x): 1F I
68. g(x) : I, - sl
3l
1
1)
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 Write the areaA of a circle as a function of
rs --ircumference C.
Cmmetry Write the arca A of an equilaterai tiangle
il"
"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.
73. 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.
1
484
2
800
3
972
4
1024
5
980
6
864
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.
;
a graphing utility to plot the points from the
::ile in part (a) with the function from part (c).
L -'.-
FL.m'closely does the function represent the data?
Erpiain.
x +24-2x-
74. Geometry A rectangular package to be sent by the
U.S. Posta] Service can have
length and girth (perimeter
a maximum
of a
combined
cross section) of
108 inches (see figure).
x
(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
to use an appropriate viewing window.
(c) What dimensions will maximize the volume of the
package? Explain.
Section
lI.
-r-r
+ !2
:
22. x2
25
:2xy -
ae.
1
sl. /(x)
4
6
f(x): [, -
1n
: [2,]
1.4
+
Graphs of
2
Functions
11
9
: [, - 2l + \
s2. f(t) : [4xn
s0.
/(x)
Function In Exercises 53 and 54, use
graphing
utility to graph the function. State the
a
domain and range of the function. Describe the pattern
Describing a Step
4
of the graph.
In
53. s(x)
-
Sketching a Piecewise-Defined Function In Exercises
55-62, sketch the graph of the piecewise-defined function
by hand.
-6
krcreasing and Decreasing Functions
Exercises
5-26, determine the open intervals on which the
hnction is increasing, decreasing, or constant.
x" .f(") :1*
2a.
f(x)
:
x2
4x
ffi
-5
-4
5./(r):x3-3xz+2
26.
f(x) : lF -
1
7
4
- ll+rll)
tr+ x/
sa. s(x) : z(1* - [*,n)'
3
-4
:2Ex
{
*
x<0
f(*):{?.
''
x>0
LJ-X.
ss.
56.
l(*)
57.
:
lx + 6. x<-4
_ +. x>-4
lz*
l./tri
[l - t*
58./{x) :l
LVx
se.
-l
4
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.
:
fi. f(*) :
!7. f(x)
3r. /(x) --
-x
30. /(x) -f/q
32. f(x) : -I - ,
28.
3
x2/3
,'G
+
Z
f(x)
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
ln. l,:2x3 + 3x2 - 12x
39. lz(x) : (* - I)-/i
AL f(x) : x2 - 4x - 5
{r. /(x) : x3 - 3x
{5."f(r):3x2 -6x* I
f(x):3xz -2x- 5
38. y: x3 - 6x2 + 15
a0. s(x) : xu@ - *
36.
f(x) : 3xz - 12x
a. f (x) : - x3 + 3x2
a2.
a6.
f(x):8x-
4x2
x<0
llx): I-;---'
LV4 - x' x20
f(x):
-
L'
+ 3.
lx -j
j3.
[2, - t.
I
[x+5'
60. g(x)
i)', x<2
x>2
x<0
0<x<2
x>2
ir : -J
: l_2. -3 < x <
[s* - +, x>
1
7
fz* + t.
6t..flx):1,
/'
lx'(^
62. h\x\: ]', I :'
L"{- T r.
x<-7
x> 1
x<0
x>0
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.
:s
6s. f(,*) : 3x - 2
,f el. n(x) : x2 - 4
6e. f(x) : Jl
: +-x
2l
7t.
63.
f(x)
f(x)
lx
6a.
f(x)
: -e
66.f(x):5-3x
68. /(-r)
: -x2 -
70. sQ):11
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.
In Exercises 47-52,
tt. (-), +)
t+. (-j,
7s. (4,e)
77. (x, -y)
76. (s, - 1)
78. (2a,2c)
a7.
l-i[6ry of
f(r)
:
fixn
+
Parent Functions
z
a8.
f(i:
[,n
-
3
8
72.f(x):-lr-51
dietch the graph of the function by hand. Then use a
graphing utility to verify the graph.
t-.
*|
-t)
120
I
Chapter
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) for several values of r.
79.f(t):t2+2t-3
80. /(x) :
./sr. g1r1 : x3 - 5x
82. h(x):
/(x)
83.
and
/(x)
85. s(s)
: x{-P
:
fQ) :
86. /(s) :
8a.
4s2/3
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
: 0.482511 0<t<8
N
- 2x2 + 3
13 - 5
t,G + s
.{6
t1.293t3
+
65.26t2
-
48.9/
+
578,
where / represents the yeal with t : 0 corresponding
to 2000. (Sonrce: Narionai Association ol Realtors)
!c3/2
Finding the lntervals Where a Function is
./ss. fuonELtNG DATA
positive In
Exercises 87-90, graph the function and determine the
E
e
interval(s) (if any) on the real axis for which
Use a graphing utility to verify your results.
87.f(x):4-x
89. /(x) : x2 - 9
:
90. .f(*) :
88.
/(x)
4r
x2
t
-
f(*) > 0.
2
4x
91. Business The cost ofusing
g
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
=
-
0.08[-(/
which years the number of cooperative homes and
condos was increasing. During which years was
- 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
Atlanta is $18.80 for
to but not including
a
I
package weighing up
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
J
Using the Graph of a
x>
0. Sketch the graph of the function.
I
t-L
t,
-i
94.
f +r -
(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.)
Function In Exercises 93 and.94,
write the height ft of the rectangle
93. l,
the number decreasing?
The cost of sending
an overnight package from New york to
:
utility to graph the model over the
(b) Use the graph from part (a) to determine during
Cr(t):1'0s+0.08[/-1n
1.0s
a graphing
appropriate domain.
of the following is the appropriate model?
czo :
(a) Use
I
as a
function of x.
{60, 100)
r00
)
4
a0
(45, s0)
e50
(.)
'.i
,(
(30. 25)
10 20 30 40 50
Time (in minutes)
60
Chapter
I
Functions and Their Graphs
Vocabulory and Concept Check
1. Name three types of rigid transformations.
2. Match
(a)
the rigid transfbrmation of
/z(,r)
:./(x) +
-r,
: /(x) with the correct
representation, where
c>
0.
(i) horizontal shift c units to the left
(ii) vertical shift c units upward
(iii) horizontal shift c units to the right
c
(b) h(x) : JQ) - ,
(c) h(:r) - .f (.' - c)
(d) /z(r) : /(.t + c)
(iv) vertical shift c units downward
In Exercises 3 and 4, fill in the blanks.
3. A reflection in the r-axis of y
:
.f(x) is represented by h(x)
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
and a vertical shrink when
when
Procedures and Problem Solving
-
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.
Sketel':ing Tr*r+s$srm*tions
5. r(x) :
s(r) :x-+
e.
"{
(e) y
: (x 2)'
s(x) -(x+2)2+2
n(i: -(-r - 2)'12. f(x) : x2
2)z
20. (a)
la. /(x)
-
:
h(x):-Jx+1
I
18. l(:r) : -
,ot:1 *z
s(r):-.r
I
s(i
- Jr
: i'G
*
2
/,(r)
4
x*3
o
i-r
(1,0)
1
- 1)
(-2,4)
f
+-i"'---"-i
I
r,L
1r')
fl -','
J
Err*r Analysis In Exercises 21 and, 22, describe the
error in graphing the function.
21.
x
I
I :./(r) -
tor .,:
-lx+31
h(*):-2lx+21 -1
16' JQ)
r\z:+
(0.
(d)y:-f(.r-2)
(e) r:l(-r')
(f y : jl("r)
lxl
s(r)
r-l
(c)r:"r(,-l)
t
i*'
-t
/z(x) : lx 3l
15. /(r) :
'6
g(r) : -,G +l
h(r):.,8-z+t
t7. f(.r) : -
.l/rz.
2/(r)
(b).y:l(x + 1)
o("1:1r2+2
+"
n(i
:
(1,2)
(0 r - l(-x)
tel ., : /(i,)
10. /(-r)
g(;r):l"rl
|
(b) v
h(*)-(x+2)2+1
- x2
:
s(x) -*+I
y-.('lx)+2
: -.f(,)
(c)v-l(r-z)
(d) r:/(x + 3)
f@: L*
s(r) :)x+2
: ,z
g(r):x2-4
e. fQ)
|
h(x\:
' x-
19. (a)
8. /(x)
h(r')-6-2)'
- -(x tt. 1Q) : "z
s(r) : jr'
h(r) : (2x)2
13. l(x) : lxl
copy of the graph, go to the website www.mcthgraphs.com.
h(*)-ia-zl
: 3,
7. f(r): 12
g(x) :x2+2
h(x)
h(.r)
Sk*t*hing Tr*nsf*rnrnti*::s In Exercises 19 and 20, use
the graph oflto sketch each graph. To print an enlarged
1(r): (r + 1)2
22.
f(x)
:
(, ]
1)'
Section 1.5
g5r"r, of Parent Functions In
F.'
Exercises 23-28,
onFare the graph of the function with the graph of its
prent function.
lx:*:.G+z
24.
:
26.
(*
!5.
.r
,F
ir,
----1-.)
|
A
-
-
y:! x -
y:
28. y :
4)'
L
s
lx + sl
-,G=a
f= tibrary of Parent Functions In Exercises 29-34,
*n@ the parent function and describe the transformation
*orn in the graph. Write an equation for the graphed
fuetion.
A
D.530.4
,r/
/(r) :
50.
s(x) :
s6) :
-\tt,)
:
h(*) f(-*)
h(x)
-
3x2
+
2
-fG)
: f(2x)
s1. s(x) :2 - (x + 5)2 s2. s@):
53. s(x):3+2(x-4)2 sa. g(x):
ss. s(x) : 3(x - 2)3
s6. s(x) :
: -(.
+
3)3
: ,Y-t6'' - e
-
-(, + 10)2 + s
-iA+T2 -2
-j(x + t)3
10
60.s(x)-x-f I-no
61. s(x) : -21* - 1l - 4 62.sG):+1,-21-3
63.sG):-i-/.+z-r 6a.g(x):--/xtt-e
se. s(r)
32.
x3
129
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/.
s8. s(ir)
-8
t
2
{+5. f(.):x3-3x2
s7.sG):@-t)3+2
-4
ltr.
Shifting, Reflecting, and Stretching Graphs
5
1
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
1
ltt.
2
34.
3
F(t)
Rigid and Nenrigid Transforrnations In Exercises
*846, compare the graph of the function with the graph
parent function.
/35.r,:-lrl
:p.r:a
t.a.
n1x)
: +lxl
{3. g(x) :
as.
"r(r)
a1x3
: -,4;
42. p(x)
:
s(.r)
h(i
!i-
x3
- 3x2
: f(x + 2)
: +i\t
20'10/used ufder llcense fr0m Shu11-"rst0ck enm
:
x3
(3, 137.3)
(e, t6t .4)
162.5)
163.5)
0q,fiZ.S)
(12.168.1)
(16, 175.0)
(10,
(11,
-
: f(* h(x) : f(3x)
s(x)
(2,132.9)
(8, 1ss.4)
\x2
y: li{
48. .f(x)
(1, 128.6)
(1,1s0.4)
c me yiewing window. Describe the graphs of
rdative to the graph of/.
f(*) :
+ 183.4.
(6, t47.4)
Rigid and Nonrigid Transfcn*aticns In Exercises 47-50,
use a graphing utility to graph the three functions in the
t7"
24.t12
(s, r43.8)
aa. y -- 2.G
+e.
*
(4, 140.8)
- l-xl
:
38. v -r:
40.v:-1x
36. y
rl" -r: (-r)2
-0.099(t
(Source: U.S. Federal Highway
Administration)
-l
dits
:
g
3x2
r)
+
and h
2
(i3,
170.0)
(1s, t14.8)
(17,
ti6.t)
(a) Describe the transformation of the parent function
f@:
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.
Section
:
.6
.l
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
-
. . S)(r)
:
71r:
+
function. look lbr an
_
and
an
function.
1). what is g(;r)'/
Cures and Pro&Jerm $o/wfngr
-; r-: :Lj:il *flTrav+ f*:r:r-ti*r:* In Exercises 7-10, frv"tl:-ratiitq +r:;+.rici:r**ire {*ml;!*+qj+* *f Sr,i*:sti*ft* In
': rraphsof/andgtographh(x)= (/+g)(-r).To Exercises 19-32, evaluate the indicated function for
' , enlargedcopyof the graph,gotothe website f(i=f
-l andg(r) =x -2algebraicalll,. Ifpossible,
- -:itqraphs.com,
use a graphing utility to verify your answer.
19, (.f + s)(3)
-l- l -*
23
-t
2t.
\f
zt.
(fg)(6)
20.
22.
za.
26.
28.
30.
32.
s)(0)
2s. (fls)es')
27.
zs.
(f
tt.
(f'lg')(-t)
dQt)
(fg)(
51)
(f
(..f
.s)(
2)
+ 8x1)
(.fg)( 4)
(fls)$)
(.f + d(.t
(/s)(3/)
(f ls)k +
4)
2')
* re ph i r:q i+ * & rit* ;-* elic il*r-s: i=i r:*ti*n *f f qJ i-.slii-=:: : In
Exercises 33-36, use a graphing utility to graph the
functions.f, g, and /z in the same viewing windou..
3
: lr. g(r) :,r - 1. ft(.r) : f(r) + g(-t)
34' f(') : 1-r, s(-t) : -r * 4. /z(r) : f(r) ,c(x)
33. l(:r)
3s. /(-r)
.: -: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. g(r)-1-x
\
.)l1l:
A \-./
r
\
5. g{-i)-.,/t -,
4. s(r) :*
-rrr
I
)
L-
-1
g(r)
-
:r3
g(.r)
36. .f(r) = 4
-.r2.
l.r. /ir.rl-/lr) 'g{.r)
g(rr - r. /rrr) :/rr)/g(,\)
-
*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(, - -
38. /(-r)
:
3e. /(x)
a0. f(r)
:
-
I
g(-r )
rl
I---
l0
:tC
l.I(-r): J-r+-:
- -3r'2- -
e(-r)
1
Chapter
'l
Functions and Their Graphs
Vocobulary ond Conceqt Check
In Exercises 1-4, frll in the blank(s)'
: x and g(/(x)) : r' then the function g is the
'
function ofl, and is denoted bY _-1, and the
of/-t is the range of f,
of-f
2. Thedomain of/is the
3. The graphs of f afif 1 are reflections of each other in the line
:J@ implies a:
4. Tohaveaninversefunction,afunction/mustbe-;thatis'/(a)
1. If/and g are functions such that/(80))
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
(*)) = x.
.f (.f '(x)) = x and/-l( f
,f t. f(x) : ex
{ s.f(r):x*7
tt. f(x) : 2x +
13. f(r) : i,6
: 1x
10. l(x):x-3
12. f(x) : (x - \)la
8.
1
t4.
f(x):
xs
its inverse function. [The graphs of the inverse functions
are labeled (a), (b), (c), and (d).1
(b)
7
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
/(r)
ldentifying Graphs of lnverse Functions In Exercises
15-18, match the graph of the function with the graph of
(a)
Verifying lnverse Functions Atgebraically
between the graPhs.
,l
: *', eQ) -- 1G 20. f(.) _- x:,
>o
21. f(t) : J* - a; s(x) : x2 + 4, x
ts.
f(*)
22.
ltrl
-- 9
: ./q -r
- 12. x > 0:
g{x)
/ zl. tQ) : t .r'. s(r) -- {
x>o; s(x):?
2a. f(r)
7
Algebraic-Graphical-Numerical
In
g(x)
:
x
o<x<1
Exercises 25-34,
are inverse functions (a) algebraically,
(b) graphically, and (c) numerically.
show
2s.
1
-1
4
(c)
4
17.
7
zs. tQ)
: - 2x*6
s(ir)
r-9
-1
4
18.
4
:
i,
sG)
fG)
: -rT-,
30. ./(x)
: 1trx -
2s.
7
I
_,1
-f,, - z,
g(x):4r-9
^
27. 1(xl-xj-l 5. g(x) :f r-5
: 16
s(r)
f(r) : 2x,
:
t12,
Q
=
g(x) =
,
x-l
,/ ll. 1t*l:
", ,
r-l-l
3{./(x) :*_,
8
16, s(x) :
!
2
32..f(x):r - 5, s(r) :
3t.
-4
:
26.flxt:
-4
t6.
f(r)
4
(d)
-4
15.
that/andg
x+
:
: - 5x-l
r- l
2xl3
P(r)-- --
R(r)
I
x<
o
Section
trar*iffingWhether Functions Have lnverses In Exercises
.G--lE. does the function haye an inyerse? Explain.
#- Dotnain Range 36.
ll2hour
I can --------- $1
6 cans -------- $5
il
cans
l-l
cans ---..--- $16
Domain
Range
51. tlxl:
<
1
Testing for One-to-One Functions In Exercises 59-70,
determine algebraically whether the function is one-toone. Verify your answer graphically. If the function is
one-to-one, find its inverse.
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.
s9.
f(x) :
60. s(-r)
40.
:
flr):
62. f(i:
6t.
42.
L
149
lx'. o <.r < I
[x. .r>
<3
58. /(x) : [t- - ']t 'r
23
4)i.
;t'
l(x -
_>$40
4hottrs/
#.
Inverse Functions
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.
lhow/-$70
2 hours -'
-$720
-$9
.tr. r-3, 6), (- 1, s), (0, 6)]
JL - rr. 4), (3,1), (1,2)j
1.7
x4
- x4
3x*4
x2
)_
3x
*5
1
63.
f6): x'
64.
h\x): x'.
4
.f(x): (a + 3)2, x>-3
66. q(x): (x s)', x<5
,f ez. f(*) : J2" + 3
6s.
6s.
/(x)
:
-2
-C
6e.fk)-lx
70. f (x) :
2
. x<2
a
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.
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.
{5./(.r)
:3-i,
x2
x']_7
h(x): \n6 -?
sl. /(r) : 10
te.
53. s(x)
s5. /z(x)
:
:
(, +
5)3
lx + 4l
l*
a6.
f(x):
lG * 2)'
r', -.
48. g(x) :- - .^
o,r"
-t
so.
/(x)
f(x): -0.65
: xs - J
-7
s2.
54. flx)
- l, - 4l
6l
lx -r ol
:
-2xJTe
,ftt.f(x):2x-3
/
f(x) -- z,
ts. fG) : *'
72.
7a.f(x):x3+1
: x3/s
: x2, ,x > 0
,/tt. Itxl: J4-=. o < x < 2
78. ltx) : J16 -?. -4 -< x < 0
4
7e. f(x) : x
75.
f(x)
76. f(x)
80. f(x)
:4
Jx
Chapter
2
Solving Equations and lnequalities
Vocabulary ond ConcePt Check
In Exercises 1-4, fill in the blank.
is a statement that two algebraic expressions are equal'
1. A(n)
2. A iinear equation in one variable is an equation that can be written in the
standard form
solution,
an equation, it is possible to introduce a(n)
which is a value that does not satisfy the original equation'
3. When solving
4.
--
Many real-life problems can be solyed using ready-made equations called
5. Is the equation x
6. How
'l 1 :
3 an identity, a conditional equation, or a contradiction?
can you clear the equation
:
1* , j
of fractions?
Procedures and Problem Solving
Checking Solutions of an Equation In Exercises 7-10,
determine whether each value of x is a solution of the
equation.
1,!:.
(a)r--;
-rr
1
I
(c)x:0
r6119
8.r*;:i
10'fr'*':
.l
+
8
3:4
1
x: i
:
Classifying Equations
In
(d)
x:
32
(a)-lr:-16
(b)x:a
16.
5l
:+-:21
xx
-
x-
(d)
'r.#
-
a conditional
;t - ro:6
tt.
,r.+-2r:f
5r-3:6-2x
2-1. 5(:-4) + 4z:5 -62
21.3r-5:2x+1
23.3(y - 5) :3 * 5r'
txr^5r1 3
,/zs.:r-;-
22.
20.
100
4rr
29.- - -
5ir
-t+
-
6
I
28. ^ -it,
r +
-
2)
-6
17*r' 3l-r :100
11'
-5.r--l l
31.--:l
1110
1a
""'.-i
-
-
.t-3
-
a1
lOx*3
5r*6
I
2
-:-
12-9
134
J+. \'-l ^T-.r r-J
12
35.-+
r r-) -:0
l
37.
1- r-r-,
3.r
2lz-41 i- l0:
1r :--rLt'
-5 - '' -
11
7)
Exercises
methods. Then
Solving Equations In Exercises 2l-10, solve the equation
(if possible).
x:16
Exercises L1-16, determine
-+:4
,r.+*t,:1
30.
2r - 2
8x + 5 : (r - ,l)2
x-l
7
0
13. -5(x - 1) : -5(.r- + 1)
14. (x + 3)(x 5) : -t' - 2(x +
15.-1-i-
x:
.t:
ll.2(x - 1):
14x
(d)
(b)
whether the equation is an identity,
equation, or a contradiction.
-
(d):r:;
1
x: -3
(c) r:21
(a)
(c)r:9
12. x2
ib,1x:4
@)r:-2 (b)x:1
(c)
t.+1
17-20, solve the equation using two
In
explain which method is easier.
Values
Equation
,,
" 2x
Solving an Equation lnvolving Fractions
6-a)(x-2)
x2',r-6
36.3-2-t
12
x-4
L-
r-2
2
1.tL
-
1
-
l0
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) = 0.
-
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
Figure for Exercises 3-6
the
x- and y-intercepts of the graph of the equation, if possible.
algebraically and graphicallY.
8.y:-1,-:
{ z.y:.t-5
Verifying Zeros of Functions In Exercises 2l-26, the
zero(s) of the function are given. Verify the zero(s) both
2
1
zt. f(x)
7
-2
22.
:
fG):
{ zl. f(*) :
Za. f(r) :
-6
y:x2+2x-l
10.v:4-x2
2
,5 /'(r'):-zJ.J\^'
Function
4(3
-
- s) + e
x3 - 6x2 + 5x
x3 - 9-r2 + 18x
3(x
r-r2
3
26.f(x):r-:-Q
,1
1
:x-E+z
11.v
12.
y: -\*-/,
+
Z
+1
4
4
-5
-5
2
4
13.y:-x+x
15. xy -2y- x* 1:0
Appro,xirnating x- and
ll.Y: 3x-l
,
16. x)-jr+4Y:0
y{nlercepts In
Exercises l7-20,
a graphing utility to graph the equation
and
approximate any .r- and y-intercepts. Verify your results
algebraically.
use
y:3(x - 2) - 6
19. y: 20 - (3x - 10)
17.
18.),:4(x+3)-2
20. y: l0 + 2(x - 2)
x:3
--')
x)
5
5
Zero(s )
r-
x:0,5,1
-lr
I
I
5
:
0, 3,6
x:1
--
.:
x
-)
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.
- 0.4x: 1.2
28.3.5x - 8:0.5;r
29. 12(x + 2) : 15(r - 4) - |
30. 1200 : 300 + 2(:r - 500)
+r(x-2) : lo 32.; r1k- 5):6
,r.;3xl-2xl
33. 0.60x + 0.40(100 - x) : 1.2
34. 0.75x + 0.2(80 - x) : 20
x-3 3x-5
- x- 3 - x-5
30.
3s.-n
,
x
27. 2.1x
r-5
.r
37.i
-;-,0
39.
40.
(x+2)':i-6x*1
("r + 1)2 + 2(x - 2) :
-^ x- 5 - .l-3
38.
ro
s
(x
+ 1)(x -
2)
:r
,
Section
2.5
207
Solving Other Types of Equations Algebraically
/ocabulory ond Concept Check
L: Erercises
L
I
and 2,
fill in the blank.
The general form of a
,1,,.\"
l.
is
+ arxl + arxl ttn:
1 0, ttr'' *''
0.
fractions. rnuitiply each side of the equation
* , : ;|
-equation
"t
To clear rhe equarion
1
by
the least common denominator
-1. Describe
the step needed to remove the radical from the equation
.t. Is the equation ra
-
2x + 4
:
-E + 2 :
*.
0 of quadratic type?
)rocedures and Problem Solving
34. 6x-7tG-3:0
:
r
36.,4+-,6-20-10
35. v[- .,6-s
37. 3"'f - 5 "( - t tz- o
38. a.[r - .t "Gi -
33.2x+9,G 5:o
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
5t-3
+
30x2
+ 45x :
-.
\.
6. 8xa -
18x2
:
-t
0
Solving an Equatian lnvolving Rational Exponents In
Exercises 39-48, find all solutions of the equation
algebraically. Check your solutions.
0
- 24x3 + 16"t2 : 0
q. t'*5:fx'+x
'rr. rf _ 2x3 : 16 * gx - ,lx3
9.ta
{
39. 3xt/3
i
+t. (, -
5)zrz
i:rving an Equetion of Quadratic Type In Exercises 43.
l1-1{, find all solutions of the equation algebraically. +S.
-heck your solutions'
Itr.,rr- 4x2 +3-o
,-1. -l,ra - 6512+ 16 : 0
- 5x2- 36:0
14. 36t4 + 29t2 - 7 :
12.
ra
0
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.
i
"a
phical
r : 13 i-. ,r - r1 -
l,<,
- 3x
10.t2 + 9
2x2
l: 2x4 - l5x3 i 1812
29x2 + 100
18. v::ra
16.
ving an Equation lnvolvlng Radieals In Exercises
-q-38, flrnd all solutions of the equation algetrraically.
,-.heck your solutions.
!:
- 10 :
:1.--10-4:O
i.r. -1,6
0
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
1-r.
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
(-r
2x2/3
-
-
5
,u
40. 9P/3
-t 24tt/t
42. (x
1):r:
-
- r,
(r, - 5x - 21trt -2 46. (r2 , 47. 3x(x - r)ttz + 2(r - r):/u - o
4g. i'.2(x_ 1)rl: +6:r(_1 1).rl: -o
16
- ,
44. (r - 7)z/: -,
9)z/t
Graphical
: -
22\+/t
:
16
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-,
_ ./15
50. v: 2r
51. r:
_ +,
- . tt "r1ar6
::;r
52. .y
- 3-rG - l
lO
-
2
Solving an Equation lnvolving Fractions In Exercises
53-64. find all solutions of the equation. Check your
solutions.
45-r
54.xJO
-::;
43
56.
1t
/ss..--*;
TI
1l)
\\ -----')
r x*1
20-x
51.
x
-
-
r-l
"\
-- l
58.4rt1:
j:t
.Yf t
1
ir
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.
-r2 : 0
, -r5, -,19-12 + 28x 1:0
5-, r.: + 3x : -8
-.q. Ji.r * 16-r'f 11 : 0
",i-1. I i
:-
.:r
2.r
e
'.,;::=
54. .r2 - 10x i 22 :
56. 9,r2 - lS_t F 9 58. -r2 + 16 : -5r
0
-
o
60.
9f -
6x
i
37
(b) Write a quadratic equation fbr the area of the floor
in terms of u.,.
(c) Find the length and wrdth of the building t1oor.
84. $esig:: arrd {*nsir=.:ti*n An above-ground swimming
O
*uc*{r*i!; iql:+tir:* In Exercises 61-68,
SolrringQuadratrrEquationsA\gebraita\ 1gl
pool with a square base is to be constructed such that the
surface area of the pool is -576 squzre f'eet. The height of
the pooi is to be 4 1'eet. (See 11gure.) What should the
dimensions of the base bel (Hitt: The surlace area is
S:12+4rh.)
solve
equation using any convenient method.
-1, r-r - 2,r I : 0
-1. I,r + 3)2 : 81
-5. ir 2t + f :0
--.
(.r
* l): - .':
62. lt;r2 - 33x: 0
64. (-r - 1)r :
1
66.,rr + 3-r j: O
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.
l'
8s. ;!4*mili-[l\d* *ATA
Ji
A gardener has 100 meters of f'encing to enciose two
2
ad.jacent rectangular gardens, as shown in the figure.
-1.
4-r+3r=100
(a) Write the area
function of
A of the enclosed
resion as
a
-r.
(b) Use a graphing utility to generate additional rows
of the table. Use the table to estimate the dimensions
thut willproduee u rrraxirrrLnl rrler.
.quations having the given solutions. (There are many
.
,
,
rrect answers.
)
-6" 5
1a
3.7
-
E
-
E
,)JJ. --\J-.J
1+2v5.
-
t, L
I
2
J.l
71. 2.1
76. l, i
78. l''5. -2.r8
80. I .r . 5. 2
82.3+11.3
I
2
l
t-eet
t8
ft:1)1
l1/
(c) Use the graphing utility to graph the
4i
of floor space.
ta) Draw a diagrarn that gives a visual representation of
l.
area function.
and use the graph to estimate the dimensions that
u i11 ploduce a maximum area.
1zl
it is wide. The buildin-s has 1632 square
the tloor: space. Represent the width as
the leneth in terms of u'.
9l
.r./S
.iyeilEte*li:r* The f-loor of a one-stor'1. buildin-e is
l-eet longer than
Area
1
and show
(dt
Use the -sraph to approrirnate the dimensions such
that the enclosed area is 3-50 square meters.
(e)
Find the required dimensions of part
algebraically.
(d)
196
2
Chapter
Solving Equations and lnequalities
Vocabulory and Concept Check
In Exercises
I
and 2,
fill in the blank.
1. An equation of the form axz * bx * c : 0, where a, b, arrd c are real numbers
and a * 0, is a _
, or a second-degree polynomial equation in r.
2. The part of the Quadratic Formula b2 - 4ac, known as the
, determines
the
type of solutions of a quadratic equation.
3. List four methods that can be used to solve a quadratic equation.
4. What does the equation s : -16t2 * vn/ * sn represent? What do vn and so
represent?
Procedures ond Problem Solving
Writing a Quadratic Equation in General Form In
Exercises 5-8, write the quadratic equation in general
form. Do not solve the equation.
{zg.*,-t4x-32:o
5.2x2:3-5"r
6.x2-25x+26
t. !(zx2 - to):
t2x
8.x(x+2)-3x7+l
Solving a Quadratic Equation by
Factoring In Exercises
9-20, solve the quadratic equation by factoring. Check
your solutions in the original equation.
9.6x2*3r-0
10.9x2-l:0
,/ ll.x2-2x-8:0
12.:,.2-10r+9:0
13. 12 + 10x + 25 :0
14.4x2il2xr9-0
15.3+5x-2x2:0
16. 2x2 - 19x + 33
17. 12 i 4x: 12
18. -x2 + 8x : 12
19. (r+ a)2-62-g
20. x2 + 2ax I a2 :0
{39.*'-4x-r13:o
40.x2-6x'f 34:0
41.x2+2x-ll'7:0
42. x2
,f
-
to
:
111
:0
6: o
26.(2r+3)2+25:0
28.
(-r
144
- s)2 :
+
5)2
-
the
43.y:(x+3)2-4
44.)':1-(x-l):
45.y:-4x2+4x-l 3
46.y:x2+3x
2s
(r +
4)2
4
Using a Graphlng Utility to Find Salutions In Exercises
47-52, use a graphing utility to determine the number of
real solutions of the quadratic equation.
- 4x -f 4:0
49.]x2 8xt28:0
16
-
+
48.2x2-x-1:0
50. jx2-5x* 25:0
47. 12
22. x2
24. (x
3)']
18r
the graph to approximate any x-intercepts of the graph,
and (c) verify your results algebraically.
nearest hundredth.
zs. (z* 1)2 +
27. (r - 7)': : (x
+
Graphing e Quadratic Equation In Exercises 43-46,
(a) use a graphing utility to graph the equation, (b) use
solutions and the decimal solutions rounded
21.
30.x2-2x-3:0
31.x2+6r*2-0
32.x2*8x*14:0
,fll.g*,-18x*3-o
34. 4x2 - 4x - 99:0
35. -6 -f 2x - x2 :0
36. -r2 + x - 1 :0
,{ 31. z*'* 5x - 8 : o
38.9x2-12x-14:0
Hxtracting Square Roots In Exercises 2l-28, solve the
equation by extracting square roots. List both the exact
x2:49
23. (x - 12)2 :
In Exercises 29-42, solve the
quadratic equation by completing the square. Verif-v
your answer graphically.
Cornpleting the Square
51. -0.2x2
-t l.2x -
8
:
52.9+2.4x-8.3x2:0
0
Section
58.
"-l
:-t
3
i
,-1.
'''
3
_
| -t
8-]i
60'
1-2i
!fi- 4-ft
- ty
ls4'"
e + 3iy
or Sub*acting Quotients of Complex Numbers
L Esercises 63-66, perform the operation and write the
rdt in standard form.
a.-
64'z+i+2-i
3
66.I +r
i
2i
sr-2i*:*s;
.
80. The product of two imaginary numbers is always
@ressions lnvolving Powers of i In Exercises 67-72,
frlify the complex number and write it in standard
lm-
G- t-/-75)'
70.
trL-i-'
72.
1
81. The conjugate of the product of two complex numbers
is equal to the product of the conjugates of the two
(J
82. The conjugate of the sum of two complex numbers is
2i3
equal to the sum of the conjugates of the two complex
numbers.
-2)6
1
_-
Qir
83. Error Analysis Describe the error.
(c) -l-Jii
Raise each complex number to the fourth power and
simplify.
ta)
r
an
imaginary number.
Cube each complex number. What do you notice?
ra)Z (b)-l+.,fli
6"
-
68. 4i2
i2
--\1
8L
an
complex numbers.
G- -6i3 +
trll
77. No complex number is equal to its complex conjugate.
imaginary number.
4-l
t
True or False? In Exercises 77-82, determine whether
the statement is true or false. Justify your answer.
78. i44 1 ;1s0 - i1a - i10e + i61 : -l
79. The sum of two imaginary numbers is always
2i.
l-i
l{?yT''"
Conclusions
ffng
)
l+l
185
Complex Numbers
I(EB],,,'o'
JGE*,,,,
olf
tf
Y, )L lr^. Y, )! ir
5i
62'
2.3
2
(b)
(c)
-2
2i
(d)
-2i
Use the results of the Explore the Concept feature on
page 182 to find each power of l.
{a) l2o (b) i4s (c') io (d) itt+
W*yyon chonld lcarn it (p. tao) The opposition to
,"** culrertt in an electrical circuit is called its
impedance. The impedance z in a parallel
circuit with two pathways satisfies the
equation l/z: 1lz, + lf zr, where zr is
the impedance (in ohms) of pathway 1, and
z, is the impedance (in ohms) of pathway 2.
Use the table to determine the impedance of
each parallel circuit. (Hint:You can find the
of each pathway in a parallel
circuit by adding the impedances of all
impedance
84. CAPSTONE Consider the binomials x
2x
-
land
the complex numbers 1
+ 5i
t 5 and
2 - i.
and
(a) Find the sum of the binomials and the sum of
the
complex numbers.
(b) Find the difference of the binomials and the
dilference o[ the complex numbers.
(c) Describe the similarities and differences in your
results for parts (a) and (b).
(d) Find the product of the binomials and the product
of the complex numbers.
the products you found in part (d) are
not related in the same way as your results in parls
1a-1 and tb,t.
(e) Explain why
(f
) Write a briel paragraph that compares operations
with binomials and ,opOiations with cornplex
numbers.
components in the pathway.)
Cu m u I otive
Resistor
adl
Symbol
Iarpedance
-::.r
a
Inductor
Capacitor
--,?l6u-
---lF
cA
ba
bi
Lugo/iStockPhoto.com
='xrov
igor
201
0/used under license flom Shutterstock.com
-cl
M ixe d Rev i ew
Multiplying Polynomials In Exercises 85-88, perform
the operation and write the result in standard form.
85. (4x - s)(a-r + s)
w. $x - l)t, *
+l
86. (x +
88. (2x
-
2)3
5)2
"t84
Chapter
2
Solving Equations and lnequalities
Vocabulary and Concept Check
1. Match the type of complex number with its definition.
(a) real number
(1) a + bi,a - 0,b + 0
(.11) a + bi, b : 0
(b) imaginary number
(c) pure imaginary number
(iii) a + bi, a * 0, b + 0
In Exercises 2 and 3, fill in the blanks.
2. The imaginary unit i is defined as I : _,
where i2 :
3. The set of real multiples of the imaginary unit i combined with
numbers is called the set of
_
the set of real
numbers, which are written in the standard
form
4. What method for multiplying two polynomials can you use when multiplying two
complex numbers?
What is the additive inverse of the complex number 2
-
6. What is the complex conjugate of the complex number
2
4i'1
-
4i?
Procedures and Problem Solving
30. ( 3.1 12.8i) (6.1 -
Equality c,f Complex Numbers In Exercises 7-10, find
real numtrers a and b such that the equation is true.
Multiplying Complex Numbers In Exercises 3l-46,
perform the operation and write the result in standard
form.
7.aIbi:-9+4i
8.4*bi:12+5i
9.(.a-1)+(r+3)i:s+81
(a+6)+2bi-6-5i
31.
10.
Writing a (omplex Number in Standard Forrn In
Exercises
ll-20, write
the complex number in standard
form.
11. s
+ J-16
12.2-J4
13. -6
15. -5r + r'
14.
17.
rc. (J-+)'z 20. -A-0004
19.
,[
0.09
t
+ i) - (1 - 2i)
22. (11 - 2i) - (-3 +
n.(-t + J-) + (s - -,T 50-)
24. (t - /- 18) + (3 - J-32)
d zs. tzi - (r4 - ti)
26. 22 + (-5 + 8l) - 9l
27. (1+ l;) + (i * *)
zs. (l + 3;) *,)
29. (t.6 + 3.2i) + (-5.8 + 4.3i)
(i
33. (J- 10)'z
35. 4(3 + s,
st. 1t + i)(3 - 2i)
39. 4l(8 + sl)
32.
34.
J=
.
"/-to
(J-is)')
36. -6(s - 3,
38. (6 2i)(2 - 3i)
40. -3i(6 - i)
(-44 + ,ani)(.m - ,40z)
42. (r5 * ,,,15r( ./1 - .asi)
43. (6 + 1i)2
44. (s (4
+ 5i)2 - (4 - 5i)2 46. (1 45.
Adding and Subtracting Complex Numbers In Exercises
2l-30, perform the addition or subtraction and write the
result in standard form.
21. (4
,f
. -/_2
J-
41.
8
16. -3i2 + i
(J -is)'
16.31)
6,)
4i)2
2i)2
-
(.t +
Zi)2
Multiplying Conjugates In Exercises 47-54, write
the complex conjugate of the complex number. Then
multiply the number by its complex conjugate.
{+t.++zi
49. e - -t5i
st. J
10
$.3-J-2
Writing
a
48.1 - 5i
50. -3 + -Di
s2. -[-13
54.1+-,/-8
Quotient of Complex Numbers in Standard Form
In Exercises 55-62, write the quotient in standard form.
6
I
t6.-,
5
Section
ii
- t'
-; i: utions of an
Equation Graphically In
,,r,.r--]so: +1-60. use a graphing utility to approximate
;m -rr-r;irrns of the equation. [Remember to write the
,L
r
1ru.irlL;rr
in the form/(x) =
$.1
42.2x3*x+4:0
:Q
ir)r+17)
44. i6.-6x+6):0
.:
+
l8r
9:0
- - ll,rr - 26-1 24 : 0
, - ._,., * I
- -+:0
,.,.l
;
l
lh
2.2
Solving Equations
Graphically
177
Finding Foints of lnterse(tion Algebraica!ly In Exercises
63-70, determine any point(s) of intersection
algebraically. Then verify your result numerically by
creating a table of values for each function.
63.
)':,
!:2r-l
64.y-7
.r
)':,
x
I
lt
2x
--1 -2r3
66.:r-_1,- 4
-rr*2r':-5
2r-tt':6
-r*'r":0
-') -l
Jf
-
- 1l :6
1
67.
'r
- )':
lo
68.4r y:4
x+2,v"-4
-t-4v:1
Expl*raticn
ra) Use
a
graphing utility
to
complete the table.
Determine the interval in which the solution to the
equation 3.2r - 5.8 : 0 is located. Explain your
reasoning.
-l
lr
3.2x
-
0
1
2
69..v:-lr2-,t*1
)':-t2+2ri1
70.
.r'- --r2+3.r*1
,.1
l-- _
I
1
3
5.8
(b) Use the graphing utility to complete the table.
Determine the interval in which the solution to the
equation 3.2x
-
5.8
:
0 is located. Explain how
this process can be used to approximate the solution
to any desired degree of accuracl,. Then use tl.re
graphing utility to verify graphical11,11. solLrtion to
3.2r-5.8:0.
1.5
"t
3.2x
62.
t.6
t.1
1.9
1.8
2
5.8
[xpl*rati**
Use the procedure in Exercise
imate the solution of the equation 0.3(x
accurate to two decimal places.
-
6l
to approx-
1.5)
- 2 :0
Apprcxirnating Points of Intersectio* Graphically In
Exercises 7l-76. use a graphing utility to approximate
an1 points of intersection of the graphs of the equations.
Check 1-our results algebraically.
72.x-3.y--2
5-t - 2r:
,/lt.t:9-2x
3
l':x
73..r':4-x2
!-2x-1
,/ls.y:zx2
l'--i'
r^l
11
74.13 .):3
.)-r,,-<
t\'
76.t:r
t,-2r
x2