Chapter 8
input.
more than one point, the graph does not
represent a function.
of days.
()
22. Domain:
{
}
6, 4, 0, 4, 6−−
28. Domain:
{
}
6, 3, 1.5,0,1, 2, 3, 4−−−
Range:
{
}
5, 2, 0,1, 4−−
32. Domain:
{}
1996, 2000, 2004, 2008
() ()
2
() ()
2
() ()
2
() ()
() ()
() ()
(
)
() ()
() ()
This cannot be simplified further without
a value for a.
148
40.
(
)
2fx x x=+
a.
() () ()
() ( )
2
222 2
24 2
82 6
f−= − +−
=+−
=−=
2
c.
() () ()
22
22fa a a a a=+=+
d.
()()()
()
()
2
2
323 3
29 3
18 3
fa a a
aa
aa
−=− +−
=+−
=−
()
b.
()
41 4g−=−
c.
()
54gn=−
d.
()
2
24gn=−
() ( )
() ( )
() ( )
() ( )
13114 1,4
030110,1
131121,2
− −−+= −
−+=
−+=− −
() ()
Domain: , ; Range: ,−∞ ∞ −∞ ∞
() ()
()
()
()
()
()
33303,0
11321,2
00330,3
11321,2
33303,0
−−−+= −
−−−+= −
−+=
−+=
−+=
() (
]
() ( )
()
()
22 2,2
02 0,2
−−
Domain:
()
,;−∞ ∞ Range: {2}
() ()
() ( )
() ( )
() ( )
2
3
44424,2
2
3
22412,1
2
3
00440,4
2
−−+=−−−
−−+=−
+=
[
)
[
)
Domain: 4, ; Range: 2,−∞ −∞
Section 8.1 Introduction to Functions
149
() ()
() ()
() ()
() ()
() ()
114551,5
004440,4
444004,0
−−−=−= −
−=−=
−= =
(
][
)
() ( )
() ( )
() ( )
2
2
010 101 0,1
111 110 1,0
−=−=
− =−=
Domain: [0, );∞ Range: ( ,1]−∞
two ordered pairs with the same first
coordinate.
Range:
{}
6,1,2, 7,33−
68.
(
)
2
3fx x x=+
() () ()
2
3
nn
=+
()
()
2
2
2
316 4
48 4
nn
nn
=+−
=−
drawn that intersects the graph at more
than one point.
72. a. Perimeter of a square is 4 times its side.
()
4ps s=
() ()
()
() ( )
() ( )
0400 0,0
12 4 12 48 12, 48
=
=
150
()
() ()
A customer would pay $230 for 4 months
of service.
c.
() ()
()
() ( )
() ( )
() ( )
45 50 ,
0 45 0 50 50 0,50
5 45 5 50 275 5,275
10 45 10 50 500 10, 500
xAx x xAx=+
+=
+=
+=
() ()
()
78. a.
() ( )
5 6000; 15 3000VV==
b. No; the width of the cardboard used to
except –1
Range: the set of all real numbers
except 1
b. Domain: the set of all real numbers
except 0
Range:
{
}
1,1−
c. Domain: the set of all real numbers
greater than –4, except 2
Range: the set of all real numbers
greater than –3, except 3
d. Domain: the set of all real numbers
Range: the set of all integers
()
evaluated.
b. Since 3 1,−≤− the rule
()
25fx x=+ is
evaluated.
() ()
() ()
32
5
=− −
=−
paired with more than one value in the range.
151
and Inverse Functions
2. The difference of f and g is defined as
4. The composition of f and g is the
6. If every horizontal line intersects the graph of
8. The graphs of a function and its inverse are
10.
() ()
36; 2fx x xgx x=− =−
()
2
36 2
xxx
=−+−
()
2
()()
()
()
322
32
36 612
312 12
xxx x
xxx
= −−+
=− +
2
()()
()()
22
2
14 5
651
xxxx
xx
=− − − −
=− − +
()()
()()
23243
4455
xxxxxx
=+−−−−
2
gxx
2
x
+
()
22
1
2
x
x
+
+
−
+
()
()
22
, 2
2
xx
x
=−
++
=− ≠−
+
()() ()
12
2
fg x x
−
⎛⎞
⋅=
⎜⎟
+
⎝⎠
x
=≠−
Chapter 8 Functions
152
()
()
16.
( )() () ()
222
fg f g+−=−+−
⎣⎦⎣⎦
18.
( )() () ()
() () ()
()( )
000
10 0 304
10 004
gf g f−=−
=−−−−
⎣⎦⎣ ⎦
=−−−−
20.
( )() () ()
() () ()
22
13141 1
⎡⎤⎡⎤
=− −−− ⋅−−
12; 3 6
()() ()
()
()
()
36
1362
3
fx
x
=−+
=−++
()() ()
()
12
3
gx
⎛⎞
=+
⎜⎟
⎝⎠
()
()
()
2
3
45 3 3
fx
xx
=+
=+ + − +
()() ()
()
()
()
2
45 3
xx
=
=+− +
()
9310
=− − − +
()
2
4
x
x
⎛⎞
=−≠−
+
b.
()() ()
()
()
6
2, 2
2
gf x gfx
gx
x
x
=
=−
=≠
−
D
2
153
36. Not one-to-one
()
The only choice that has the point
()
5, 3−−
()
()
The only choice that has the points
()
1,1 and
()
4,2 is choice b.
42.
()()()
1: 2, 9 , 3, 5 , 4, 2 ,
f−−−−
⎧⎫
46.
()
()
7
7 Substitute for .
7 Interchange and .
fx x
yx y fx
xy x y
=
=
=
7
1 Interchange and .
50.
()
()
10
10 Substitute for .
10 Interchange and .
fx x
yx y fx
xy x y
=+
=+
=+
()
3
11 Interchange and .
3
1
13
31
xy x y
xy
xy
=+
−=
−=
1 Interchange and .
3
x
xxy
y
=−
58.
()
32
322
223
fx x
xy x y
xyxy
=+
+=
=−
154
()
()
()
27
277
fx
x
x
=−
−+
=
()() ()
()
7
2
7
gf x gfx
x
g
x
=
+
⎛⎞
=⎜⎟
⎝⎠
+
⎛⎞
D
()
14
3
fx
⎛⎞
=+
⎜⎟
⎝⎠
( )() ()
()
()
()
()()
312
13124
3
gx
x
=+
=++
()() ()()
fgx x gf x≠≠
DD
()
7
6
x
p
−
⎛⎞
=⎜⎟
⎝⎠
( )() ()
()
()
()
67
767
6
76 7
qx
x
x
=−+
−− +
=
+−
=
()() ()
()
58
15
fx
x
⎛⎞
=−+
⎜⎟
⎝⎠
=⋅
=
()() ()
()
58
5
gf x gfx
=
=− +
−
⎛⎞
D
155
68.
() ()
5,0 and 0, 1 are on the function.
−−
()()()
()
and 2,4 are on the inverse function.
−
()()
()
()
2
2
257 2
257 2
fgx x x x
xx x
+=−++−+
=−+−+
()
2
2
257 2
245
xx x
xx
=−++−
=−+
()()
()
()
2
322
32
257 2
24510714
2 9 17 14
fg x x x x
xxx xx
xx x
⋅=−+−+
=−++−−+
=− + − +
()() ()
2,0 , 0, 1 , and 2, 2 are on the
−− −
()
17 Substitute for .
4
17 Interchange and .
4
yx yfx
xy xy
=−
=−
()
b.
()() ()
()
()
0.5 8 70
Sx h Sxh
h
=
D
()
()
1
6254
6 254
6
254
6
yh
hy
h
wh
−
=−
=−
+
=
b. The inverse can be used to determine
the height of a man for a given ideal
body weight.
156
84. a.
()
175 5000
5000 175
5000
175
5000
At t
ty
ty
t
=+
−=
−=
−
invested for a given account balance.
Mindstretchers
() ()
20 0 000 000
0 4 2 422426
133 330336
20 4 044044
35 55510550
−+=−=
−−+=−−−=−
−−+=−−=−
+= −=−
+= −=
() ()
() ()
()
()
()
()
()
32
61
hx f g x
fgx
gx
x
=
=
=+
=−
D
()
()
()
()
3261
363
63
gx x
gx x
x
+= −
=−
−
() ()
x
()
()
11
1
ff x
−=
D
()
()
The function is an inverse of itself.
b.
() ()
1
gx x g x
−
=− =
()
() ( )
1
gg x x
−=− −
D
()
()
The the function is an inverse of itself.
() ()
x
()
()
12
2
hh x
x
−=−
−
D
()
()
The function is an inverse of itself.