Chapter 8
FUNCTIONS
8.1 Introduction to Functions
Exercises
2. A function’s first coordinates are values of
the independent variable.
10. Yes; for each year, there is a single
number of smart phones sold.
12. No; some months have the same number
of days.
14. Not a function. The points
()
1,1 and
()
1, 1− have the same first element
(as do
()
2, 4− and
()
2, 4 ).
20. The relation is a function.
22. Domain:
{
}
6, 4, 0, 4, 6−−
Range:
{
}
3, 2, 0, 2, 3−−
{
}
30. Domain:
{
}
0,1, 2, 3
Range:
{
}
9
32. Domain:
{}
1996, 2000, 2004, 2008
Range: 45,590,703, 50,456,062,
62,040,610, 69, 498,459
⎧⎫
⎪⎪
⎪⎪
⎨⎬
⎪⎪
⎪⎪
⎩⎭
d.
()()
1
5.2 5.2 1 2.6 1 3.6
2
g−=−−=−−=−
36.
(
)
3.5 1.5fx x=− +
a.
() ()
3 3.5 3 1.5 10.5 1.5 9f=− + =− + =−
b.
() ()
1 3.5 1 1.5 3.5 1.5 5f−=− −+=+=
c.
() ()
3.5 1.5 3.5 1.5fn n n=− + =− +
33 3
110
110 9
=−−
=− =−
b. 11
12
22
h⎛⎞ ⎛⎞
=−
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
Chapter 8 Functions
148
40.
(
)
2
2fx x x=+
a.
() () ()
() ( )
2
222 2
24 2
f−= − +−
=+−
42.
(
)
4gx=−
a.
()
23 4g=−
44.
() ( )
() ( )
() ( )
() ( )
31 ,
13114 1,4
030110,1
131121,2
xfx x xy=− +
− −−+= −
−+=
−+=− −
46.
() ()
()
()
()
()
()
3,
33303,0
11321,2
00330,3
11321,2
33303,0
xgx x xy=− +
−−−+= −
−−−+= −
−+=
−+=
−+=
46. (continued)
()
22 2,2
Domain:
()
,;−∞ ∞ Range: {2}
50.
() ()
34,
2
3
xfx x xy=+
[
)
[
)
Domain: 4, ; Range: 2,−∞ −∞
Section 8.1 Introduction to Functions
149
52.
() ()
() ()
4,
114551,5
xgxx xy=−
−−−=−=−
54.
() ( )
() ( )
() ( )
() ( )
2
2
2
2
1,
010 101 0,1
111 110 1,0
21 2 14 3 2,3
xfx x xy=−
−=−=
− =−=
−=−=− −
64. Not a function:
()
0,3 and
()
0,5 are
two ordered pairs with the same first
d.
()()()
()
()
2
2
2
434 4
316 4
48 4
fn n n
nn
nn
−=− +−
=+−
=−
70. Not a function. A vertical line can be
drawn that intersects the graph at more
than one point.
72. a. Perimeter of a square is 4 times its side.
()
4ps s=
b.
() ()
12 4 12 48,p== so the perimeter is
Chapter 8 Functions
150
74. a.
()
45 50Ax x=+
b.
() ()
4 45 4 50 180 50 230A=+=+=
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
xAx x xAx=+
+=
+=
76. a.
()
1.2 10,000Cx x=+
b.
() ()
500 1.2 500 10,000
600 10,000 10,600
C=+
=+ =
The manufacturing plant’s monthly costs
total $10,600 if 500 units are produced.
c.
() ()
()
1.20 10,000 ,
xCx x xCx=+
Mindstretchers
1. a. Domain: the set of all real numbers
except –1
Range: the set of all real numbers
except 1
b. Domain: the set of all real numbers
except 0
Range:
{
}
1,1−
evaluated.
c. i. 41−≤−
() ()
4245
85
3
f−= −+
=− +
=−
ii. 11−≤−
() ()
1215
f−= −+
Section 8.2 The Algebra of Functions and Inverse Functions
151
8.2 The Algebra of Functions
and Inverse Functions
Exercises
2. The difference of f and g is defined as
()()()()
.fgx fx gx−=−
10.
() ()
2
36; 2fx x xgx x=− =−
()()
()
()
2
2
2
36 2
36 2
352
fgx x x x
xxx
xx
+=−+−
=−+−
=−−
()()
()
()
2
2
2
36 2
36 2
372
fgx x x x
xxx
xx
−=−−−
=−−+
=−+
()()
()
()
2
322
32
36 2
36612
31212
fg x x x x
xxx x
xxx
⋅=−−
= −−+
=− +
()
()
2
36
2
32
2
3, 2
fxx
x
gx
xx
x
xx
⎛⎞ −
=
⎜⎟ −
⎝⎠
−
=−
=≠
12. (continued)
()()
()()
22
23243
432
14 5
4455
593
fgx xxxx
xxxxxx
xxxx
⋅=−− +
=+−−−−
=− − − +
()
()( )
2
2
14 5
115
fxx
x
gxx
xx
⎛⎞ −−
=
⎜⎟ +
⎝⎠
+−
=+
12 4
2
23
, 2
2
x
x
xx
x
−+ +
=+
+
=≠−
+
()()
()
()
()
12
2
22
1
22
12 4
2
25
, 2
2
fgx x
x
xx
x
x
xx
x
−
−=−
+
+
−
=−
++
−− −
=+
−−
=− ≠−
+
()() ()
12
2
2, 2
2
fg x x
x
x
−
⎛⎞
⋅=
⎜⎟
+
⎝⎠
−
=≠−
+
()
1
2
2
fx
x
g
−
⎛⎞ +
=
⎜⎟
⎝⎠
Chapter 8 Functions
152
For exercises 16−22,
()
234fx x x=−− and
()
2
1.gx x=−
18.
( )() () ()
() () ()
()( )
22
000
10 0 304
10 004
14 5
gf g f−=−
⎡⎤⎡ ⎤
=−−−−
⎣⎦⎣ ⎦
=−−−−
=+ =
22.
()
() ()
()
2
2
4344
4
14
16 12 4
116
24 8
15 5
f
g
−−−−
⎛⎞
−=
⎜⎟ −−
⎝⎠
+−
=−
==−
−
b.
()() ()
()
12
3
1
326
3
66
gf x gfx
gx
x
x
x
=
⎛⎞
=+
⎜⎟
⎝⎠
⎛⎞
=− + +
⎜⎟
⎝⎠
=− − +
=−
D
26.
() ()
2
45 ; 3fx x x gx x=+ − =+
a.
()() ()
()
()
3
fgx fgx
fx
=
=+
D
b.
()() ()
()
()
()
2
2
2
45
45 3
57
gf x gfx
gxx
xx
xx
=
=+−
=+− +
=− + +
D
d.
( )() () ()
()
33537
953 7
9157
13
gf =− + +
=− + +
=− + +
=
D
28.
() ()
2
6; 4
fx x gx x
=− =+
a.
()() ()
()
fgx fgx
=
D
()
64
2, 2
2
x
x
x
=≠
−
c.
()()
()
2
26
24
26
2
16
fg−= −
−+
=−
=−
D
Section 8.2 The Algebra of Functions and Inverse Functions
153
30. One-to-one
32. Not one-to-one
40. The function has the points
()
1,1 and
()
2, 4 .
The only choice that has the points
()
1,1 and
()
4,2 is choice b.
48.
()
()
()
1
1
7
1 Substitute for .
7
1 Interchange and .
7
7
7
gx x
yx ygx
xy xy
xy
gx x
−
=
=
=
=
=
52.
()
()
49
4 9 Substitute for .
4 9 Interchange and .
hx x
yx yhx
xy x y
=−
=−
=−
()
11 Substitute for .
3
11 Interchange and .
3
1
13
yx yfx
xy x y
xy
=+
=+
−=
()
1
31
x
gx x
−
+
=
58.
()
()
2
32
2 Substitute for .
32
2 Interchange and .
32
x
fx x
x
yyfx
x
y
xxy
y
=+
=+
=+
Chapter 8 Functions
154
60.
() ()
7;27
2
x
fx gx x
+
==−
( )() ()
()
()
27
fgx fgx
fx
=
=−
D
62.
() ()
1
312; 4
3
fx x gx x=+ = +
()() ()
()
14
3
1
3412
3
fgx fgx
fx
x
=
⎛⎞
=+
⎜⎟
⎝⎠
⎛⎞
=++
⎜⎟
⎝⎠
D
64.
() ()
7
67; 6
x
px x qx −
=− + =
( )() ()
()
7
pq x pqx
x
=
−
⎛⎞
D
66.
() ()
55
;8
8
fx gx
xx
−
==−+
−
()() ()
()
58
5
588
fgx fgx
fx
=
⎛⎞
=−+
⎜⎟
⎝⎠
−
=
D
Section 8.2 The Algebra of Functions and Inverse Functions
155
68.
() ()
() ()
5,0 and 0, 1 are on the function.
0, 5 and 1,0 are on the inverse
function.
−−
−−
70.
()()() ()
()()()
()
4, 2 , 1,1 , 1, 1 , and 4, 2 are
on the function. 2, 4 , 1, 1 , 1,1 ,
and 2, 4 are on the inverse function.
−− − −
−−−
−
72.
(
)
(
)
22
352; 8fx x x gx x=−+ =−
( )()() () ()
[]()()
()( )
22
33833532
98 39 5 3 2
98 27152
144 43
gf ⎡⎤⎡ ⎤
−−=−−−−−−+
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦
⎡⎤
=−− −−+
⎣⎦
=−− ++
=− =−
76.
()() ()
()() ()
2,0 , 0, 1 , and 2, 2 are on the
function. 0, 2 , 1,0 , and 2,2
are on the inverse function.
−− −
−− −
78.
()
()
17
4
17 Substitute for .
4
17 Interchange and .
fx x
yx yfx
=−
=−
b.
()() ()
()
()
()
8
0.5 8 70
470
Sx h Sxh
Sh
h
h
=
=
=+
=+
D
This represents the employee’s daily
salary for h hr of work.
c.
( )() ()
7.5 4 7.5 70 30 70 100Sx =+=+=D
The employee’s daily salary is $100.
Chapter 8 Functions
156
84. a.
()
175 5000
175 5000
175 5000
At t
yt
ty
=+
=+
=+
invested for a given account balance.
Mindstretchers
1.
() () ()
()()
()
()()
() ()
35 15 145 16
hx kx
xfxgx fgx fgx
==
+−
−−+−=−−=
2.
() ()
61, 32hx x f x x=− =+
() ()
()
()
()
hx f g x
fgx
=
=
D
3. Answers may vary.
Possible answers are given.
1
1
−
()
()
The function is an inverse of itself.
b.
() ()
1
gx x g x
−
=− =
()
() ( )
1
gg x x
x
−=− −
=
D
()
()
()
()
11
gg x x g g x
−−
==DD
2
12
x
x
x
⎛⎞
=− ⋅ −
⎜⎟
⎝⎠
=
()
()
()
()
11
hh x x h h x
−−
==DD