Chapter 8
Basics of Functions
8.1 Check Points
1. The domain is the set of all first components. The
domain is {0, 10, 20, 30, 40}.
b. The relation is a function.
3. a. () 4 5
(6) 4(6) 5
(6) 29
fx x
f
f
=+
=+
=
b.
2
2
() 3 10
(5) 3(5) 10
(5) 65
gx x
g
g
=−
−=− −
−=
4. a. Every element in the domain corresponds to
exactly one element in the range.
b. The domain is {0, 1, 2, 3, 4}.
The range is {3, 0, 1, 2}.
8.1 Concept and Vocabulary Check
1. relation; domain; range
8.1 Exercise Set
1. The relation is a function.
The domain is {1, 3, 5}.
3. The relation is not a function.
The domain is {3, 4}.
The range is {4, 5}.
4. The relation is not a function.
The domain is {5, 6}.
The range is {6, 7}.
5. The relation is a function.
The domain is {-3, -2, -1, 0}.
The range is {-3, -2, -1, 0}.
The range is {1}.
9. a.
(0) 0 1 1f=+=
b.
(5) 5 1 6f=+=
e.
()
(2) 21
21 3
fa a
aa
+=++
=++=+
10.
() 3fx x=+
Chapter 8 Basics of Functions
c.
(8) 8 3 5f−=−+=−
11. a.
()
(0) 3 0 2 0 2 2g=−=−=−
b.
() ()
5352
15 2 17
g−=−−
=− − =−
12.
() 4 3gx x=−
a.
() ()
040303 3g=−=−=−
b.
() ()
5453 203 23g−= −−=−−=−
13. a.
() ()
2
(0) 3 0 5 3 0 5
055
h=+=+
=+=
e.
()
22
(4)3453165
hb b b
=+= +
a.
() () ()
020 4204
04 4
h=−=−
=−=−
b.
() () ()
2
121 4214
24 2
h−= − −= −
=−=−
e.
() ()
()
2
525 4225 4
50 4
hb b b
b
=−= −
=−
15. a.
() ()
2
(0) 2 0 3 0 1
001 1
f=+−
=+−=−
2
32 12 1 19
=−−=
d.
() ()
2
2
() 2 3 1
231
fb b b
bb
=+−
=+−
Section 8.1 Introduction to Functions
16.
2
() 3 4 2fx x x=+−
a.
() () ()
()
2
030 402
30 0 2
002 2
f=+−
=+−
=+−=−
d.
() () ()
2
2
342
342
fb b b
bb
=+−
=+−
17. a.
32
(0) ( 0) (0) (0) 7
7
f=− − − +
=
b.
32
(2) ( 2) (2) (2) 7
7
f=− − − +
=−
18. a.
32
(0) ( 0) (0) (0) 10
10
f=− − − +
=
b.
32
(2) ( 2) (2) (2) 10
4
f=− − − +
=−
Chapter 8 Basics of Functions
19. a.
()
()
20 3 03
(0) 04 04
33
44
f−−
==
−−
−
==
−
d.
()
()
25 3 10 3
(5) 54 9
13 13
99
f−− −−
−= =
−− −
−
==
−
20.
31
() 5
x
fx x
−
=−
a.
() ()
()
30 1 01 1 1
005 5 55
f−−−
====
−−−
e.
()
()
31
5
331
5
ah
fa h ah
ah
ah
+−
+= +−
+−
=+−
b.
(3) 16f=
c.
0x=
23. a.
(2) 2h−=
25.
() ()
()
()
()() ()
2
131535 2
12224
42410
g
fg f
=−=−=−
=−=−−−+
=++=
Section 8.1 Introduction to Functions
29.
() ()
()()
33
33
3
55
55
22
fx fx
xx xx
xx xx
xx
−−
=− +− − − + −
=− − − − − +
=− −
31. a.
() ()
2325 65 1f−=−+=−+=−
b.
() ()
0407077f=+=+=
32. a.
() ()
3631 181 19f−= −−=−−=−
b.
() ()
0703033f=+=+=
33. a. {(Iceland, 9.7), (Finland, 9.6), (New Zealand, 9.6),
(Denmark, 9.5)}
34. a. {(Bangladesh, 1.7), (Chad, 1.7), (Haiti, 1.8),
(Myanmar, 1.8)}
b. Yes, the relation is a function because each
country in the domain corresponds to exactly
one corruption rating in the range.
35. – 38. Answers will vary.
39. makes sense
43. false; Changes to make the statement true will vary.
A sample change is: All functions are relations.
44. false; Changes to make the statement true will vary.
A sample change is: Functions can have ordered
48. false; (4) (4)
(1) (1)
2
gf−+ −
=− +−
=−
Chapter 8 Basics of Functions
49. ()3()7337
() 3 7
fa h a h a h
fa a
+= ++= ++
=+
50. Answers will vary. An example is {(1, 1), (2, 1)}.
51. It is given that ()()()fx y fx fy+= +
and (1) 3f=.
To find (2)f, rewrite 2 as 1 + 1.
(2) (1 1) (1) (1)
ff ff=+= +
52.
()
2
24 4 2 5 2 6
÷−− −
53.
222
22 2 5 10
3524
33
39
xy x y y
yyxx
−−
−
===
54. 34
35
xx
=+
59 9960
460
460
44
15
xx xx
x
x
x
−=−+
−=
−=
−−
=−
The solution set is
15−
.
1. a. The graph represents a function. It passes the
vertical line test.
b. When x is 9, the function’s value is 100. i.e.
(9) 100f=
c. The minimum T cell count during the
asymptomatic stage is approximately 425.
Section 8.2 Graphs of Functions
4. a. The domain is [2,1]−.
The range is [0,3] .
8.2 Concept and Vocabulary Check
1. ordered pairs
2. more than once; function
{
}
8.2 Exercise Set
1. The graph represents a function. It passes the
vertical line test.
2. The graph represents a function. It passes the
vertical line test.
{
}
6. The graph does not represent a function. It fails the
vertical line test.
7. The graph does not represent a function. It fails the
vertical line test.
12.
(4) 4f−=
17.
()
10 2g−=
18.
(10) 2g=−
19. When
()
2, 1.xgx=− =
22.
{
}
24xx−<≤
23.
{
}
52xx−≤ <
26.
{
}
25xx−≤≤
Chapter 8 Basics of Functions
29.
{
}
3xx≥−
{
}
{
}
32.
{
}
2xx<
35. The domain is [0,5) .
The range is [1,5)−.
36. The domain is (5,0]−.
The range is [3,3)−.
39. The domain is [2,6]−.
The range is [2,6]−.
40. The domain is [3,2]−.
The range is [5,5]−.
43. The domain is
{
}
5, 2, 0,1, 3−− .
The range is
{
}
2.
c.
(3) 4f−=
d. 2 and 6; i.e. (2) (6) 2ff==−
46. a. The domain is (,6].−∞
b. The range is (,1].−∞
c.
(4) 1f−=−
d. –6 and 6; i.e. (6) (6) 3ff−= =−
h.
(2)f− is positive.
47. a.
( ) 0.013 56.46
(80) 0.013(80) 56.46
57.5
fx x
f
=+
=+
=
Section 8.2 Graphs of Functions
48. a.
( ) 0.013 56.46
(90) 0.013(90) 56.46
fx x
f
=+
=+
49. a.
( ) 1.27 305
(50) 1.27(50) 305
368.5
fx x
f
=+
=+
=
b.
2
2
( ) 1.01 0.65 310
(50) 0.01(50) 0.65(50) 310
367.5
gx x x
g
=++
=++
=
50. a.
( ) 1.27 305
(40) 1.27(40) 305
355.8
fx x
f
=+
=+
=
The average carbon dioxide concentration in
1990 was 355.8 parts per million.
This is represented by the point
(40, 355.8).
51.
() ()
2
(20) 0.420 3620 1000
f=−+
52.
() ()
2
(50) 0.4 50 36 50 1000
f=−+
() ()
()
(45) 0.4 45 36 45 1000
0.4 2025 1620 1000
810 1620 1000
f=−+
=−+
=− +
for any driver between ages 16 and 74.
54. Answers will vary.
One possible answer is age 16 and age 74.
()
0.4 5476 2664 1000
2190.4 2664 1000 526.4
=−+
=−+=
Both 16-year-olds and 74-year-olds have
approximately 526.4 accidents per 50 million miles
driven.
55. (3) 0.78f=
The cost of mailing a first-class letter weighing 3
$0.61.
Chapter 8 Basics of Functions
62. The number of physician’s visits per year
based on age first decreases and then
increases over a person’s lifetime.
63. makes sense
64. makes sense
65. does not make sense; Explanations will vary.
Sample explanation: The domain is the set of
67. false; Changes to make the statement true will vary.
A sample change is: The graph of a vertical line is
not a function.
68. true
A sample change is: (0) 0.6f=
73.
2
2
(1.5) (0.9) [ ()] (3) (1) ( )
1 0 [ 4] 2 ( 2)(3)
ff ffff
ππ
−+−− +−÷ ⋅−
=+−−+÷−
74.
2
2
(2.5) (1.9) [ ( )] (3) (1) ()
2 ( 2) [3] 2 ( 2)( 4)
49(1)(4)
294
fff fff
ππ
−− −− +−÷ ⋅
=−−− +÷−−
=−+−−
=−+
3
2
10 15
10 15
10 10
x
x
x
−=−
−−
=
−−
=
3
624 2(3 8) 2
624 6 16 2
624 8 16
8608
76
xx
xx
x
x
x
=++
=++
=+
−=−
=
79.
2
(4) (4) (4 4) (4 5)
20 ( 1)
fg+=++−
=+−
Section 8.3 The Algebra of Functions
8.3 Check Points
1. a. The function contains neither division nor a square root. For every real number, x, the algebraic expression
1
2
3x+
is
2. a.
()
()
()
2
() () ()
34127
fgx fx gx
xx x
+=+
=+−++
3. a.
()
57
() 8
fgx xx
−=−
−
b. The domain of f – g is the set of all real numbers that are common to the domain of f and the domain of g. Thus, we
must find the domains of f and g.
Note that
5
()fx x
=
is a function involving division. Because division by 0 is undefined, x cannot equal 0.
4. a.
( )() () ()
2
5 5 5 [5 2 5] [5 3] 23fg f g+=+=−⋅++=
b.
( )() () ()
22
[2][3] 33fgx fx gx x x x x x−=−=−−+=−−
Chapter 8 Basics of Functions
5. a.
( )() () ()
22
22
( 2.6 49 3994) ( 0.6 7 2412)
2.6 49 3994 0.6 7 2412
BDx Bx Dx
xx xx
xx xx
+=+
=− + + +− + +
=− + + − + +
8.3 Concept and Vocabulary Check
1. zero
2. negative
7. (,)−∞ ∞
8. (2, )∞
9. (0,3) ; (3, )∞
8.3 Exercise Set
1. Domain of f is (,).−∞ ∞
Section 8.3 The Algebra of Functions
4. Domain of g is ( , 5) or ( 5, )−∞ − − ∞ .
5. Domain of f is (,3)or(3,)−∞ ∞ .
11.
()()( )
() 3 1 2 6
3126
55
fgx x x
xx
x
+=++−
=++−
=−
() ()
(5) 5 5 5
25 5 20
fg+=−
=−=
13.
()()
()
2
2
2
() 5 3
53
35
fgx x x
xx
xx
+=−+
=−+
=+−
()()()
()
2
525 56
225 5 6
50 5 6 49
fg+=+−
=+−
=+−=
15.
()
()
()
2
()
231
fgx
xx x
+
=−−++
2
2
431
42
xx x
x
=−−++
=−
( )() () ( )
2
545 24252
100 2 98
fg+=−=−
=−=
17.
()()( )
() 5 2 3
fgx x x
+=+−−
73
x
=+
() ()( )
2
() 5 2 3
10 15
fg x x x
xx
=−−
=− −
435
5
xx
x
=− + −
=− −
() ( )( )
2
() 4 3 5
fg x x x
=− − +
Chapter 8 Basics of Functions
19. Domain of =( , ).fg+−∞∞
20. Domain of =( , ).fg+−∞∞
26. Domain of = =( , 8) or ( 8,4) or (4, ).fg fg++−∞−− ∞
27. Domain of =( ,2) or (2, ).fg+−∞ ∞
31.
2
2
( )() () ()
42
32
fgx fx gx
xx x
xx
+=+
=++−
=++
() ()
2
()(3)333220fg+ =++=
35.
()
()
2
( )() () ()
42
fgx fx gx
xx x
−=−
=+−−
Section 8.3 The Algebra of Functions
36.
()
()
2
2
( )() () ()
42
42
fgx fx gx
xx x
xx x
−=−
=+−−
=+−+
39.
2
( )() () ()
(4)(2)
fg x f x g x
xx x
=⋅
=+ −
40.
2
32
( )() () ()
(4)(2)
28
fg x f x g x
xx x
xxx
=⋅
=+ −
=− − +
32
32
()() 2 8
(3) 2(3) 8(3)
27 18 24
21
fg x x x x=− − +
=− − +
=− − +
=−
Chapter 8 Basics of Functions
44.
() ()
()
2
4
2
fx
fxx
x
ggx x
+
==
−
() () ()
2
343
912 21
321
23 1 1
f
g
+
+
====−
−−−
46.
() ()
()
24
2
fx
fxx
x
ggx x
+
==
−
() () ()
2
040
00 0
00
20 2 2
f
g
+
+
====
−
50.
( )() () ()
()
()
242fg x f x g x x x x=⋅=+ −
Domain of
(,)fg =−∞∞
.
51.
( )() () ()
333415fg f g+−=−+−=+=
52.
( )() () ()
222231gf g f− −=−−−=−=−
53.
( )() () () ( )()
222111fg f g==−=−
57. The graph of
fg+
59.
()()()()
() () ( ) ( )
11
11[1 1]
fg gf
fgg f
+−−−
=+−−−−
61.
()() ()
()() ()
()
2
2
2
2
21
1
22 1
6
50 02 04 4
3
f
fg g
f
fg g
−−
=− −−
−
=⋅− =− =−=−
−
63. a.
( )() () ()
(1.5 115) (1.4 121)
2.9 236
MFx Mx Fx
xx
x
+=+
=+++
=+
64. a.
()
()()()
(1.4 121) (1.5 115)
0.1 6
FMx Fx Mx
xx
x
−=−
=+−+
=− +
65. a.
( ) 1.5 115
() ( ) 1.4 121
MMxx
x
FFxx
+
==
+
b. 1.5 115
() 1.4 121
Mx
x
Fx
+
=
+
66. First, find
()()
R
Cx−.
( )( ) 65 (600,000 45 )
65 600,000 45
R
Cx x x
xx
−=− +
=− −
()
( )(30, 000)
20 30,000 600,000
600, 000 600,000 0
RC−
=−
=−=
If the company produces and sells 30,000 radios, it
67. – 70. Answers will vary.
71.
12312
2 3 2 2 yx y xyyy=+ =− =+
72.
12
312
4 2yx y x
yyy
=− =
=−
Chapter 8 Basics of Functions
75.
76. makes sense
77. makes sense
78. makes sense
83. false; Changes to make the statement true will vary.
A sample change is:
()fa
or
()fb
is 0.
84.
()
3
33
33
Rab
Rab
Ra b
=+
=+
−=
86.
()()
26 24
612468
fb b
bb
+= +−
=+−=+
88.
2
3222
3
xxx
−
+=−+=
Mid-Chapter Check Point – Chapter 8
1. The relation is not a function.
The domain is
{1, 2} .
The range is
{6,4,6}.−
The range is [0,3].
4. The relation is not a function.
The domain is
(3,4].−
The range is [1,2].−
exactly one element in the range. It passes the
vertical line test.
8.
()
43f−=
Section 8.4 Composite and Inverse Functions
12. The range of
f
is
(,4].−∞
15.
() ()
() ()
() ( )
2
003088
10 2 10 5 20 5 15
01081523
f
g
fg
=− +=
− =− − −= −=
+− =+=
18.
()()
2
2
3825
53
fgx x x x
xx
+=−++−−
=−+
( )()() ()
2
22523
410317
fg+−=−−−+
=+ +=
36
=−
2
38
16 12 8 36 12
85 3
fxx
−+
++
===
−
22. The domain of
f
g
is
55
,or,
22
−∞ − − ∞
.
b.
()() ()
()
()
2
2
56
(5 6) 1
25 60 36 1
gf x gfx
gx
x
xx
=
=+
=+−
=++−
Chapter 8 Basics of Functions
4.
()
27
27
fx x
yx
=+
=+
5. a. Since a horizontal line can be drawn, it fails the
horizontal line that intersects the graph more
than once test. Thus, this graph does not
represent a function that has an inverse function.
6. Since f has a line segment from
(2,2)−−
to
(1,0),−
then
1
f
−
has a line segment from
(2,2)−−
to
(0, 1).−
Since f has a line segment from
(1,0)−
to
(1, 2),
8.4 Concept and Vocabulary Check
1. composition;
(())fgx
6. false
7. inverse
8.4 Exercise Set
1. a.
()() ()
()
()
7
fgx fgx
fx
=
=+
2. a.
()() ()
()
()
()
5
35315
fgx fgx fx
xx
==−
=−=−
b.
()() ()
()
()
335gf x gfx g x x===−
()
()
4
241
28129
gx
x
xx
=+
=++
=++=+
c.
2225459fg =+=+=