College Algebra
Content
Math 1314
College Algebra
Fall 2010
LSC – North Harris
Latest Update
October 30, 2012
Email Comments
Material taken from:
Weltman, Perez, Tiballi
unpublished material
“College Algebra” version
7
3
by Stitz and Zeager
Go to LSC‐North Harris Math Department
website for updated and corrected
versions of this material.
Math Dept Website: nhmath.lonestar.edu
Page 1
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Page 13
Page 14
x
-6-5-4-3-2-1 1 2 3 4 5 6
x
-6-5-4-3-2-1 1 2 3 4 5 6
x
-6 -5 -4 -3 -2 -1 1 2 3 4
x
-6-5-4-3-2-1 1 2 3 4 5 6 7 8 910
x
-1 1 2 3 4 5 6 7 8 91011
x
-1
-
2
3
-
1
3
1
3
2
3
1
x
2 7
3
8
3
3 10
3
11
3
4
Section 2.4-Absolute Value
1. 8 or 8 = = ÷ x x 3. 6 or 6 = = ÷ x x 5.
5 5
or
4 4
= = ÷ x x
7. 4 or 3 = = ÷ x x 9. 10 or 2 = = ÷ x x 11.
7
or 2
2
= = ÷ x x
13.
8
2 or
3
÷
= ÷ = x x 15. No solution 17.
10 14
or
3 3
= = ÷ x x
19.
8
8 or
3
= = ÷ x x 21.
5
2
= ÷ x 23.
14 8
or
5 5
= = x x
25.
4 2
or
15 3
= = ÷ x x 27. 5 or 4 = = ÷ x x 29.
10
2 or
3
= ÷ = ÷ x x
31.
1
3
= x 33. No solution 35.
17 13
or
12 12
÷
= = x x
37. 3 or 1 = = ÷ x x 39. 2 or 4 = = x x 41.
11
2
= ÷ x
43.
5 1
or
3 5
= = x x 45.
11
3 or
7
= = ÷ x x
51.
( )
55 ÷ , 53.
( )
1 1 ÷· ÷ · , ( , )
55. 4 2 ÷ [ , ] 57. 59 ÷ [ , ]
59.
(
11 1 ÷· ÷ · (
¸
, [ , ) 61.
(
5 7 ÷· · (
¸
, [ , )
63.
( )
5 4 ÷ , 65.
1
3
= x
67. No Solution 69.
11
3
3
| |
|
\ .
,
Page 15
Answers
x
-2 -1 1 2 3 4
x
-2
-
3
2
-1
-
1
2
1
2
1 3
2
2
x
-4 -2 2 4
2/3
x
1 2 3 4 5 6 7 8 91011
x
-3
-
8
3
-
7
3
-2
-
5
3
-
4
3
-1
-
2
3
-
1
3
1
3
2
3
1
x
-3-2-1 1 2 3 4 5 6 7 8
x
-3 -2 -1 1 2 3
x
-10-9 -8 -7 -6 -5 -4 -3 -2 -1
x
-2 -1 1 2 3 4 5 6 7
x
-5 -4 -3 -2 -1 1 2
x
-3 -2 -1 1 2 3
x
-6-5-4-3-2-1 1 2 3 4 5 6
71.
1
3
3
| |
÷· ÷ ·
|
\ .
, ( , )
73.
1 1 1
or
2 2 2
| | | |
= ÷ ÷· ÷ ÷ ·
| |
\ . \ .
, ,
x
75.
( )
÷· · , 77.
(
2
2
3
, ,
|
÷· ÷ · (
¸ |
¸ .
79.
( )
48 , 81.
7
1
3
÷ | |
|
\ .
,
83.
(
1 5 , [ , ) ÷· ÷ · (
¸
85.
9 7
4 4
, ,
| ( |
÷· ÷ ·
| (
\ ¸ ¸ .
87.
( )
9 1 ÷ ÷ , 89.
( )
15 ÷ ,
91.
10 2
3 3
, ,
| ( |
÷· ÷ ·
| (
\ ¸ ¸ .
93.
( )
0 1 ÷· · , ( , )
Page 16
Absolute Value Inequalities
Solve the inequalities. Graph your answer on a number line. Write answers
in interval notation.
1. 3 x ≤
2. 5 x <
3. 2 7 x + ≤
4. 3 5 10 x − <
5.
2
4
3
x −
<
6. 2 3 10 x + ≤
7. 4 2 3 9 x + − <
8. 3 1 7 10 x − + ≤
9. 1 4 7 2 x − − < −
10. 1 2 4 1 x − − ≤ −
11. 4 x ≥
12. 7 x >
13. 1 8 x + ≥
14. 2 1 7 x − ≥
15.
1
3
5
x −
>
16. 3 4 8 x + ≥
17. 2 10 16 x + ≥
18. 3 1 2 4 x − − >
19. 9 2 2 1 x − − ≥ −
20. 2 4 5 8 x + − >
21. 2 3 12 x − <
22. 2 5 9 x + ≥
23. 2 3 5 13 x − + ≤
24. 4 1 7 13 x − − >
25. 2 1 12 5 x + + =
26. 4 5 4 x − − =
27. 2 5 6 x − − ≤
28. 1 0
2
x
+ ≥
29. 6 7 0 x + ≤
30. 3 6 2 > + + x
31. 4 8 6 ≤ + + x
32. 1 0 x + >
33. 5 5 1 3 < + + x
34. 5 9 7 4 < + − x
Page 17
1-10
21-34
11-20
Absolute Value Inequalities- Answers
1.
| |
3,3 −
2.
( ) 5,5 −
3.
| |
9,5 −
4.
5
,5
3
| |
−
|
\ .
5.
( )
10,14 −
6.
13 7
,
2 2
(
−
(
¸ ¸
7.
( )
5,1 −
8.
| |
0,2
9.
3
1,
2
| |
−
|
\ .
10.
| |
1,2 −
11.
( | | ) , 4 4, −∞ − ∞ ∪
12.
( ) ( )
, 7 7, −∞ − ∞ ∪
13.
( | | )
, 9 7, −∞ − ∞ ∪
14.
( | | ) , 3 4, −∞ − ∞ ∪
15.
( ) ( )
, 14 16, −∞ − ∞ ∪
16. ( |
4
, 4 ,
3
|
−∞ − ∞
|
¸ .
∪
17.
( | | )
, 18 2, −∞ − − ∞ ∪
18.
( ) ( ) , 1 3, −∞ − ∞ ∪
19.
( | | )
,4 5, −∞ ∞ ∪
20.
21 5
, ,
2 2
| | | |
−∞ − ∞
| |
\ . \ .
∪
21.
9 15
,
2 2
| |
−
|
\ .
22.
( | | ) , 7 2, −∞ − ∞ ∪
23.
| |
1,7 −
24.
( ) ( )
, 4 6, −∞ − ∞ ∪
25. ∅
26. 5 x =
27.
( ) , −∞ ∞
28.
( )
, −∞ ∞
29.
6
7
x = −
30.
( )
, −∞ ∞
31. ∅
32.
( ) ( )
, 1 1, −∞ − − ∞ ∪
33. ∅
34. ∅
Page 18
Page 19
Page 20
Page 21
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
Factoring Review
Page 32
Page 33
Section 2.5—Quadratic Equations
1.
1
, 5
3
−
3.
1 3
,
2 2
− −
5.
1
, 0
4
7. { } 2, 2 −
9. { } 1, 4
11. { } 4, 5 −
13.
3
2
15. { } 7, 3
17.
{ }
3 ±
19.
3 5
5
±
21.
11
2
±
23.
{ }
3 7 ±
25. { } 2,1 −
27.
3 6
2
− ±
29.
{ }
2 2 ±
31.
1
3
i
− ±
33.
{ }
2 2 ±
35. { } 6, 2 − −
37.
{ }
3 10 ±
39. { } 5 3i − ±
41.
5 13
2
− ±
43.
2 3
2
− ±
45.
5 57
4
±
47.
1 59
6
i
− ±
49.
{ }
1 3 ±
51. { } 4 i − ±
53.
5
2
55.
{ }
3 5 i ±
57.
7
0,
2
−
59.
3
2
i − ±
61.
2 5
3
±
63.
{ }
4 2 ±
65.
4
, 0
3
67.
2
2
±
69.
1 5
3
i − ±
71.
3 7
7
±
73.
1 2
3
i
±
75.
1
, 3
2
−
−
77.
1
2
i − ±
Page 34
Answers
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Section 2.7—Miscellaneous Equations
1.
2
0, , 3
5
−
3. { } 0, 5, 2 − −
5.
3
,1, 1
2
−
−
7.
2 2
2, ,
3 3
−
9.
1
, 2
3
i
±
11.
1 3
3,1,
2 2
i
−
− ±
13.
1
2, ,1 3
2
i
− ±
15.
5
2
−
17. ∅
19. { } 4
21. { } 9
23. { } 1 −
25. { } 6
27. { } 1 −
29. { } 32
31. { } 3
33. { } 1 −
35. { } 3
37.
3
, 2
3
± ±
39.
3
,
2
i
± ±
41.
{ }
2 13 ± ±
43.
7 3
2
− ±
±
45.
1
, 8
27
−
47.
27 1
,
8 8
− −
49.
1
, 4
4
51. { } 9
53.
2
, 3
3
−
55.
1
, 2
4
57.
9 15
,
4 4
59.
1 3
,
3 5
−
61. { } 7, 3 −
63.
1
, 2
2
65. { } 1,15 −
67. { } 16,11 −
69. { } 31, 33 −
71.
80
3
Page 46
Answers
Quadratic Types of Equations
Find all solutions of the equation.
1.
4 2
13 40 0 x x − + =
2.
4 2
5 4 0 x x − + =
3.
6 3
2 3 0 x x − − =
4.
6 3
7 8 0 x x + − =
5.
2 1
3 3
3 2 5 x x + =
6.
4 2
3 3
5 6 0 x x − + =
7.
1 1
2 4
2 1 0 x x − + =
8.
1 1
2 4
4 4 0 x x − + =
9.
1 1
2 4
4 9 2 0 x x − + =
10.
1 1
2 4
3 2 0 x x − + =
11.
2 1
3 3
2 5 3 0 x x − − =
12.
2 1
3 3
3 5 2 0 x x + − =
13.
4 2
3 3
4 65 16 0 x x − + =
14.
2 1
10 24 0 x x
− −
− − =
15.
2 1
3 7 6 0 x x
− −
− − =
16.
2 1
2 7 4 0 x x
− −
− − =
17.
2 1
7 19 6 x x
− −
+ =
18.
2 1
5 43 18 x x
− −
− =
19.
2 1
6 2 x x
− −
+ =
20.
4 2
9 35 4 0 x x
− −
− − =
21. ( ) ( )
2
2 7 2 12 0 x x + + + + =
22. ( ) ( )
2
2 5 2 5 6 0 x x + − + − =
23. ( ) ( )
2
3 4 6 3 4 9 0 x x + − + + =
24. ( ) ( )
2
2 2 20 0 x x − + − − =
25. ( ) ( )
2
2 1 5 1 3 x x + − + =
26.
( ) ( )
1 1
2 4
2 11 2 18 x x − = − −
Page 47
Answers
Quadratic Types of Equations
1. 2 2, 5 x = ± ±
2. 2, 1 x = ± ±
3.
3
3, 1 x = −
4. 2,1 x = −
5.
125
,1
27
x = −
6. 2 2, 3 3 x = ± ±
7. 1 x =
8. 16 x =
9.
1
,16
256
x =
10. 16,1 x =
11.
1
,27
8
x = −
12.
1
, 8
27
x = −
13.
1
, 64
8
x = ± ±
14.
2 5
,
3 8
x = −
15.
3 1
,
2 3
x = −
16.
1
2,
4
x = −
17.
7 1
,
2 3
x = −
18.
5 1
,
2 9
x = −
19.
3
2,
2
x = −
20.
1
3 ,
2
x i = ± ±
21. 5, 6 x = − −
22.
7
1,
2
x = − −
23.
1
3
x = −
24. 7, 2 x = −
25.
3
2,
2
x = −
26. 6563,18 x =
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Page 57
Page 58
Page 59
Page 60
Page 61
-2
-
5
3
-
4
3
-1
-
2
3
-
1
3
1
3
2
3
1
-1 1 2 3 4 5 6 7
-2 -1 1 2 3 4 5 6
-1
-
1
2
1
2
1
2 – 3 2 + 3
–4 – 2 –4 + 2
-5 -4 -3 -2 -1 1 2 3 4 5
-7 -6-5-4-3-2-1 1 2 3 4
-1 1 2 3 4 -2 -1 1 2 3 4
-1
-
1
2
1
2
1 3
2
2 5
2
3 -7 -6-5-4-3-2-1 1 2 3 4
-4 -3 -2 -1 1 2 1 2 3 4 5 6
Section 2.8-Polynomial and Rational Inequalities
1.
1
2
3
(
÷
(
¸ ¸
, 3.
( )
4 5 ÷· · , ( , )
5.
9
0
2
| (
÷· ÷ ·
(
\ ¸
, [ , ) 7.
( )
15 ÷ ,
9.
1
2
= x 11.
( )
2 3 2 3 ÷ + ,
13.
( ) ( )
4 2 4 2 ÷· ÷ ÷ + · , ,
15.
( )
÷· · ,
17. No Solution 19.
( )
5 32 ÷· ÷ ÷ , ( , )
21.
( )
1 23 3 ÷· · , ( , ) ( , )
23.
)
1 ÷ ·
¸
,
25.
1 3
0
2 2
| | | |
·
| |
\ . \ .
, ,
27.
( ) ( )
5 33 ÷· ÷ ÷ , ,
29.
)
3 ÷ ·
¸
, 31.
( )
25 ,
Page 62
Answers
-2 -1 1 2 3 4 5 6 7
-4
-
7
2
-3
-
5
2
-2
-
3
2
-1
-
1
2
1
2
1
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-8 -7 -6 -5 -4
-3 -2 -1 1 2 3 4 5
-1 1 2 3 4 5 6
-10 -8 -6 -4 -2 2 4 6
-8-7-6-5-4-3-2-1 1 2 3 4 5 6
-6 -4 -2 2
-8 -7 -6 -5 -4 -3 -2 -1 1 2
33.
(
1 6 ÷· ÷ · (
¸
, ( , )
35.
( )
1
3
2
÷ | |
÷· ÷ ·
|
\ .
, ,
37.
)
65 ÷
¸
, 39.
( )
6 ÷· ÷ ,
41.
( )
2 4 ÷ , 43.
(
2 4 ÷· · (
¸
, ( , )
45.
( )
10 4 ÷· ÷ · , ( , )
47.
(
6 14 ÷· ÷ ÷ (
¸
, ( , )
49.
( )
4 2 ÷· ÷ ÷ · , ( , )
51.
( )
13
41
2
÷ | |
÷· ÷
|
\ .
, ,
Page 63
Page 64
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Page 86
Section 3.1-Relations and Functions
1.
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
A
B
C
D
3. 10, (3, 1) d M = = ÷
5.
3
17, ( ,1)
2
d M = =
7. 2 29, (3,3) d M = =
9.
5 1
7 2, ,
2 2
d M
÷ | |
= =
|
\ .
11.
109 11 1
, ,
12 8 4
d M
| |
= =
|
\ .
13. 23.05 4.8, ( 0.15, 2) d M = ~ = ÷ ÷
15.
5
3 13, , 2
2
d M
| |
÷
= =
|
|
\ .
17.
1
1
2
, ,
2
x h
d h M y
+ | |
= =
|
\ .
19. (10,11) B =
21.
11
( ,6)
2
23. 10 6 5, 20 P A = + =
27. (7,2);( 9,2) ÷
29. ( 3,12);( 3,2) ÷ ÷
31. (0,9);(0, 7) ÷
37. ( 3,0); 37 Center radius = ÷ =
Page 87
Answers
Page 88
Page 89
Page 90
Page 91
Page 92
Page 93
Page 94
Page 95
Page 96
Page 97
Page 98
In exercises 25-42 an equation and its graph are given. Find the intercepts of the graph, and
determine whether the graph is symmetric with respect to the x-axis, y-axis, and/or the origin.
25. 26. 27.
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3
4
5
6
y =x
2
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3
4
5
6
y =x
4
y =3– |x|
28. 29. 30.
x =y
2
+2
y =2x y =–3x
31. 32. 33.
x
2
+y
2
=4 4x
2
+9y
2
=36
y =
1
x
Page 99
34. 35. 36.
1
x
2
+1
x =|y| +3
y = 4– x
37. 38. 39.
x
2
– y
2
=1
y
2
– x
2
=1
x =y
2
– 4
40. 41. 42.
y = 4– x
2
y =x
2
+2
x = 4– y
2
Page 100
Page 101
In exercises 69-72, sketch a graph that is symmetric to the given graph with respect to the y-axis.
69. 70.
71. 72.
In exercises 73-76, sketch a graph that is symmetric to the given graph with respect to the x-axis.
73. 74.
Page 102
75. 76.
In exercises 77-80, sketch a graph that is symmetric to the given graph with respect to the origin.
77. 78.
79. 80.
Write Algebra
81. Explain what it means for the graph of an equation to be symmetric with respect to the
y-axis, x-axis, or the origin.
82. How do you find the intercepts of the graph of an equation?
83. Describe a strategy for finding the graph of an equation.
Page 103
Section 3.2-Graphs of Equations
1. 3. 5.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-12
-9
-6
-3
3
7. 9. 11.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
13. 15. 17.
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-6 -5 -4 -3 -2 -1 1 2
-4
-3
-2
-1
1
2
3
4
Page 104
19. 21. 23.
-4-3-2-1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
-6-5-4-3-2-1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
25. (0,0);symmetric with respect to y-axis
27. (0,3),( 3,0),(3,0); symmetric with respect
to y-axis
29. (0,0);symmetric with respect to origin
31. (0, 2),( 2,0),(2,0); symmetric with respect to
y-axis, x-axis, and origin
33. No intercepts; symmetric with respect to origin
35. (3,0);symmetric with respect to x-axis
37. ( 1,0); symmetric with respect to y-axis, x-axis,
and origin
39. (0, 2),( 4,0); symmetric with respect to x-axis
41. (0,2);symmetric with respect to y-axis
43. 45. 47.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
-4 -3 -2 -1 1 2 3 4
-1
1
2
3
4
5
6
49. 51. 53.
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
7
8
Page 105
55. 57. 59.
-2 -1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
7
8
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2-1 1 2 3 4 5 6 7 8 910
-4
-3
-2
-1
1
2
3
4
61. 63. 65.
-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
67. 69. 71.
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
73.
75.
77.
79.
Page 106
Page 107
Page 108
Page 109
Page 110
Page 111
Page 112
Page 113
Page 114
Page 115
Page 116
Page 117
Page 118
Section 3.4-Relations and Functions
1. Yes, it is a function
3. No, not a function
5. Yes, it is a function
7. Yes, it is a function
9.
3 5
0 4
2 10
1 3 1
2
3
2
3 3
3
3
÷ = ÷
=
=
÷ = +
÷
=
÷
÷ + ÷ ÷
=
+ ÷
=
( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
f
f
f
f x x
f x f
x
f h f
h
f x h f x
h
11.
2
3 13
0 5
2 3
1 2 4 3
2
2 4
2
3 3
2 12
4 2
÷ = ÷
=
= ÷
÷ = ÷ + +
÷
= ÷ ÷
÷
÷ + ÷ ÷
= ÷ +
+ ÷
= ÷ ÷
( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
f
f
f
f x x x
f x f
x
x
f h f
h
h
f x h f x
x h
h
13.
2
3 10
0 7
2 5
1 2 10
2
6
2
3 3
2
2 4
÷ = ÷
= ÷
=
÷ = + ÷
÷
= +
÷
÷ + ÷ ÷
= ÷
+ ÷
= + +
( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
f
f
f
f x x x
f x f
x
x
f h f
h
h
f x h f x
x h
h
15.
2
3 15
0 9
2 5
1 3 5 7
2
3 7
2
3 3
3 17
6 3 1
÷ = ÷
=
= ÷
÷ = ÷ + +
÷
= ÷ ÷
÷
÷ + ÷ ÷
= ÷ +
+ ÷
= ÷ ÷ ÷
( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
f
f
f
f x x x
f x f
x
x
f h f
h
h
f x h f x
x h
h
17.
5
3
3
0
5
2
2
5
1
1
2 5
2 2
3 3 5
3 3
5
÷ = ÷
=
=
÷ =
÷
÷
= ÷
÷
÷ + ÷ ÷
=
÷ +
+ ÷ ÷
=
+
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
f
f undefined
f
f x
x
f x f
x x
f h f
h h
f x h f x
h x x h
19.
1
3
6
1
0
3
2 1
1
1
4
2 1
2 3
3 3 1
6 6
1
3 3
÷ = ÷
= ÷
= ÷
÷ =
÷
÷
=
÷ ÷
÷ + ÷ ÷
=
÷ +
+ ÷ ÷
=
÷ + ÷
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( )( )
f
f
f
f x
x
f x f
x x
f h f
h h
f x h f x
h x x h
Page 119
Answers
21.
3 6
0 0
2
2
3
2 2
1
3
2 4
2 3 4
3 3 8
1
8
4 4
÷ = ÷
=
=
÷
÷ =
+
÷
=
÷ +
÷ + ÷ ÷
=
+
+ ÷
=
+ + +
( )
( )
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )( )
f
f
f
x
f x
x
f x f
x x
f h f
h h
f x h f x
h x x h
23.
( )
2 2 ÷· ÷ ÷ ·
, ( , )
25.
( )
÷· · ,
27.
5
2
| (
÷·
(
\ ¸
,
29. 2 5 ÷· ÷ ·
( , ] [ , )
31.
( )
÷· · ,
33.
)
2 ·
¸
,
35.
( )
÷· · ,
37.
( )
3 34 4 ÷· ·
, ( , ) ( , )
39.
( )
÷· · ,
41.
( )
÷· · ,
43.
) ( )
03 3·
¸
, ,
45.
( )
5· ,
Page 120
50 Relations and Functions
1.5.1 Exercises
1. Suppose f is a function that takes a real number x and performs the following three steps in
the order given: (1) square root; (2) subtract 13; (3) make the quantity the denominator of
a fraction with numerator 4. Find an expression for f(x) and find its domain.
2. Suppose g is a function that takes a real number x and performs the following three steps in
the order given: (1) subtract 13; (2) square root; (3) make the quantity the denominator of
a fraction with numerator 4. Find an expression for g(x) and find its domain.
3. Suppose h is a function that takes a real number x and performs the following three steps in
the order given: (1) square root; (2) make the quantity the denominator of a fraction with
numerator 4; (3) subtract 13. Find an expression for h(x) and find its domain.
4. Suppose k is a function that takes a real number x and performs the following three steps in
the order given: (1) make the quantity the denominator of a fraction with numerator 4; (2)
square root; (3) subtract 13. Find an expression for k(x) and find its domain.
5. For f(x) = x
2
−3x + 2, find and simplify the following:
(a) f(3)
(b) f(−1)
(c) f
3
2
(d) f(4x)
(e) 4f(x)
(f) f(−x)
(g) f(x −4)
(h) f(x) −4
(i) f
x
2
6. Repeat Exercise 5 above for f(x) =
2
x
3
7. Let f(x) = 3x
2
+ 3x −2. Find and simplify the following:
(a) f(2)
(b) f(−2)
(c) f(2a)
(d) 2f(a)
(e) f(a + 2)
(f) f(a) + f(2)
(g) f
2
a
(h)
f(a)
2
(i) f(a + h)
8. Let f(x) =
x + 5, x ≤ −3
√
9 −x
2
, −3 < x ≤ 3
−x + 5, x > 3
(a) f(−4)
(b) f(−3)
(c) f(3)
(d) f(3.001)
(e) f(−3.001)
(f) f(2)
Page 121
Additional Exercises taken from Stitz and Zeager Book
Suggested problems are p. 50: 5-10
1.5 Function Notation 51
9. Let f(x) =
x
2
if x ≤ −1
√
1 −x
2
if −1 < x ≤ 1
x if x > 1
Compute the following function values.
(a) f(4)
(b) f(−3)
(c) f(1)
(d) f(0)
(e) f(−1)
(f) f(−0.999)
10. Find the (implied) domain of the function.
(a) f(x) = x
4
−13x
3
+ 56x
2
−19
(b) f(x) = x
2
+ 4
(c) f(x) =
x + 4
x
2
−36
(d) f(x) =
√
6x −2
(e) f(x) =
6
√
6x −2
(f) f(x) =
3
√
6x −2
(g) f(x) =
6
4 −
√
6x −2
(h) f(x) =
√
6x −2
x
2
−36
(i) f(x) =
3
√
6x −2
x
2
+ 36
(j) s(t) =
t
t −8
(k) Q(r) =
√
r
r −8
(l) b(θ) =
θ
√
θ −8
(m) α(y) =
3
y
y −8
(n) A(x) =
√
x −7 +
√
9 −x
(o) g(v) =
1
4 −
1
v
2
(p) u(w) =
w −8
5 −
√
w
11. The population of Sasquatch in Portage County can be modeled by the function P(t) =
150t
t + 15
, where t = 0 represents the year 1803. What is the applied domain of P? What range
“makes sense” for this function? What does P(0) represent? What does P(205) represent?
12. Recall that the integers is the set of numbers Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}.
8
The
greatest integer of x, x, is defined to be the largest integer k with k ≤ x.
(a) Find 0.785, 117, −2.001, and π + 6
(b) Discuss with your classmates how x may be described as a piece-wise defined function.
HINT: There are infinitely many pieces!
(c) Is a +b = a +b always true? What if a or b is an integer? Test some values, make
a conjecture, and explain your result.
8
The use of the letter Z for the integers is ostensibly because the German word zahlen means ‘to count.’
Page 122
1.5 Function Notation 53
1.5.2 Answers
1. f(x) =
4
√
x −13
Domain: [0, 169) ∪ (169, ∞)
2. g(x) =
4
√
x −13
Domain: (13, ∞)
3. h(x) =
4
√
x
−13
Domain: (0, ∞)
4. k(x) =
4
x
−13
Domain: (0, ∞)
5. (a) 2
(b) 6
(c) −
1
4
(d) 16x
2
−12x + 2
(e) 4x
2
−12x + 8
(f) x
2
+ 3x + 2
(g) x
2
−11x + 30
(h) x
2
−3x −2
(i) x
4
−3x
2
+ 2
6. (a)
2
27
(b) −2
(c)
16
27
(d)
1
32x
3
(e)
8
x
3
(f) −
2
x
3
(g)
2
(x −4)
3
=
2
x
3
−12x
2
+ 48x −64
(h)
2
x
3
−4 =
2 −4x
3
x
3
(i)
2
x
6
7. (a) 16
(b) 4
(c) 12a
2
+ 6a −2
(d) 6a
2
+ 6a −4
(e) 3a
2
+ 15a + 16
(f) 3a
2
+ 3a + 14
(g)
12
a
2
+
6
a
−2
(h)
3a
2
2
+
3a
2
−1
(i) 3a
2
+ 6ah + 3h
2
+ 3a + 3h −2
8. (a) f(−4) = 1
(b) f(−3) = 2
(c) f(3) = 0
(d) f(3.001) = 1.999
(e) f(−3.001) = 1.999
(f) f(2) =
√
5
9. (a) f(4) = 4
(b) f(−3) = 9
(c) f(1) = 0
(d) f(0) = 1
(e) f(−1) = 1
(f) f(−0.999) ≈ 0.0447101778
Page 123
54 Relations and Functions
10. (a) (−∞, ∞)
(b) (−∞, ∞)
(c) (−∞, −6) ∪ (−6, 6) ∪ (6, ∞)
(d)
1
3
, ∞
(e)
1
3
, ∞
(f) (−∞, ∞)
(g)
1
3
, 3
∪ (3, ∞)
(h)
1
3
, 6
∪ (6, ∞)
(i) (−∞, ∞)
(j) (−∞, 8) ∪ (8, ∞)
(k) [0, 8) ∪ (8, ∞)
(l) (8, ∞)
(m) (−∞, 8) ∪ (8, ∞)
(n) [7, 9]
(o)
−∞, −
1
2
∪
−
1
2
, 0
∪
0,
1
2
∪
1
2
, ∞
(p) [0, 25) ∪ (25, ∞)
11. The applied domain of P is [0, ∞). The range is some subset of the natural numbers because
we cannot have fractional Sasquatch. This was a bit of a trick question and we’ll address the
notion of mathematical modeling more thoroughly in later chapters. P(0) = 0 means that
there were no Sasquatch in Portage County in 1803. P(205) ≈ 139.77 would mean there were
139 or 140 Sasquatch in Portage County in 2008.
12. (a) 0.785 = 0, 117 = 117, −2.001 = −3, and π + 6 = 9
Page 124
1.6 Function Arithmetic 61
4. Find and simplify the difference quotient
f(x + h) −f(x)
h
for the following functions.
(a) f(x) = 2x −5
(b) f(x) = −3x + 5
(c) f(x) = 6
(d) f(x) = 3x
2
−x
(e) f(x) = −x
2
+ 2x −1
(f) f(x) = x
3
+ 1
(g) f(x) =
2
x
(h) f(x) =
3
1 −x
(i) f(x) =
x
x −9
(j) f(x) =
√
x
3
(k) f(x) = mx + b where m = 0
(l) f(x) = ax
2
+ bx + c where a = 0
3
Rationalize the numerator. It won’t look ‘simplified’ per se, but work through until you can cancel the ‘h’.
Page 125
Taken from Stitz and Zeager
1.6 Function Arithmetic 63
4. (a) 2
(b) −3
(c) 0
(d) 6x + 3h −1
(e) −2x −h + 2
(f) 3x
2
+ 3xh + h
2
(g) −
2
x(x + h)
(h)
3
(1 −x −h)(1 −x)
(i)
−9
(x −9)(x + h −9)
(j)
1
√
x + h +
√
x
(k) m
(l) 2ax + ah + b
Page 126
ANSWERS
Domain of a Function
Find the domain of the following (write answers in interval
notation):
1.
2
2
( )
5 6
x
f x
x x
=
+ +
2.
2
( )
9
x
f x
x
=
−
3.
2
3 7
( )
6 27
x
f x
x x
+
=
− −
4.
3 2
8
( )
8 2 3
x
f x
x x x
+
=
− −
5.
2
4
( )
25
x
f x
x
=
−
6. ( ) 3 5 f x x = −
7. ( ) 5 f x x = +
8. ( ) 3 7 f x x = −
9. ( ) 12 24 f x x = −
10. ( ) 9 27 f x x = − +
11.
5
( ) 2 f x x = −
12.
1
( )
3 1
f x
x
=
+
13.
12
( )
5
x
f x
x
=
−
14.
2
5 7
( )
9
x
f x
x
+
=
−
15.
2
( ) 4 f x x = −
16.
2
( ) 12 11 5 f x x x = + −
17.
2
( ) 5 6 f x x x = + +
18. 4 3 ) (
2
− − = x x x f
19.
2
( ) 2 8 f x x x = − −
20.
2
( ) 9 f x x = −
21.
2
( ) 100 f x x = −
22. 4 ) (
2
+ = x x f
23.
2
( ) 6 12 f x x x = − −
24.
2
( ) 15 4 3 f x x x = − −
25.
2
5
( )
121
x
f x
x
−
=
−
26.
4
( ) 3 15 f x x = −
Page 127
Domain of a Function-Answers
1.
( ) ( ) ( )
, 3 3, 2 2, −∞ − − − − ∞ ∪ ∪
2.
( ) ( ) ( ) , 3 3,3 3, −∞ − − ∞ ∪ ∪
3.
( ) ( ) ( )
, 3 3,9 9, −∞ − − ∞ ∪ ∪
4.
1 1 3 3
, ,0 0, ,
2 2 4 4
| | | | | | | |
−∞ − − ∞
| | | |
\ . \ . \ . \ .
∪ ∪ ∪
5.
( ) ( ) ( )
, 5 5,5 5, −∞ − − ∞ ∪ ∪
6.
( )
, −∞ ∞
7. [ 5, ) − ∞
8.
7
,
3
|
∞
|
¸ .
9. [2, ) ∞
10.
( |
,3 −∞
11.
( )
, −∞ ∞
12.
1
,
3
| |
− ∞
|
\ .
13.
( )
5,∞
14. ( ) ( ) , 3 3, −∞ − ∞ ∪
15.
( | | )
, 2 2, −∞ − ∞ ∪
16.
5 1
, ,
4 3
| ( |
−∞ − ∞
|
(
\ ¸ ¸ .
∪
17.
( | | )
, 3 2, −∞ − − ∞ ∪
18.
( | | ) , 1 4, −∞ − ∞ ∪
19.
( | | )
, 2 4, −∞ − ∞ ∪
20.
| |
3,3 −
21.
| |
10,10 −
22.
( )
, −∞ ∞
23.
4 3
, ,
3 2
| ( |
−∞ − ∞
|
(
\ ¸ ¸ .
∪
24.
1 3
, ,
3 5
| ( |
−∞ − ∞
|
(
\ ¸ ¸ .
∪
25.
| ) ( ) 5,11 11,∞ ∪
26. [5, ) ∞
Page 128
Page 129
Page 130
Page 131
Page 132
Page 133
Page 134
Page 135
Page 136
Page 137
Page 138
Page 139
Page 140
In exercises 21-38 determine the domain and range of each functions whose graph is given.
Express your answers using interval notation.
21. 22. 23.
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
24. 25. 26.
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
27. 28. 29.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Page 141
30. 31. 32.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3
33. 34. 35.
-4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
36. 37. 38.
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
Page 142
In exercises 39-40, use the graphs to determine the intervals where each function is increasing,
decreasing, or constant. Express your answers using interval notation.
39. 40.
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
G(x)
Sketch the following piecewise functions:
41.
2 1 0
( )
5 0
x if x
f x
x if x
+ ≤ ⎧
=
⎨
− >
⎩
42.
3 0
( )
2 0
x if x
f x
x if x
− − ≤ ⎧
=
⎨
+ >
⎩
43.
1 2
( )
3 9 2
x if x
f x
x if x
+ ≤ ⎧
=
⎨
− + >
⎩
44.
3 2
( )
3 5 2
x if x
f x
x if x
− ≤ ⎧
=
⎨
− >
⎩
45.
2
3 1
( )
1
x if x
f x
x if x
≤ ⎧
⎪
=
⎨
> ⎪
⎩
46.
4 2
( )
2 2
if x
f x
x if x
≤ − ⎧
=
⎨
− > −
⎩
47.
3 5 2
( )
1 2
x if x
f x
if x
+ ≤ − ⎧
=
⎨
− > −
⎩
48.
4 1
( )
1 1
x if x
f x
if x
+ ≠ ⎧
=
⎨
− =
⎩
49.
2 1 2
( )
5 2
x if x
f x
if x
− ≠ ⎧
=
⎨
=
⎩
50.
2
0
( )
3 0
x if x
f x
if x
⎧
≠ ⎪
=
⎨
= ⎪
⎩
51.
2
1
( )
4 1
x if x
f x
if x
⎧ ≠
=
⎨
=
⎩
52.
1
5 2
2
( ) 3 7 2 3
1 3
x if x
f x x if x
x if x
⎧
+ < −
⎪
⎪
= − + − ≤ ≤
⎨
⎪
+ >
⎪
⎩
53.
6 4
1
( ) 7 4 2
2
5 2
x if x
f x x if x
x if x
+ < − ⎧
⎪
⎪
= + − ≤ ≤
⎨
⎪
− + > ⎪
⎩
54.
2 5 0
( ) 5 0 4
2 4
x if x
f x if x
x if x
− + < ⎧
⎪
= ≤ <
⎨
⎪
− ≥
⎩
55.
1 1
( ) 1 1
1 1
if x
f x x if x
if x
− < − ⎧
⎪
= − ≤ <
⎨
⎪
≥
⎩
56.
1 2
( ) 3 2 3
2 9 3
x if x
f x if x
x if x
− + ≤ − ⎧
⎪
= − < <
⎨
⎪
− + ≥
⎩
57.
2 4 1
( ) 2 1 2
4 2
x if x
f x if x
x if x
+ ≤ − ⎧
⎪
= − < <
⎨
⎪
− + ≥
⎩
-6 -4 -2 2 4 6
-4
-2
2
4
6
F(x)
Page 143
Section 3.5-Interpreting Graphs
1. 3
3. 3
5. 1
7. 2 −
9. 4 −
11. Yes, it is a function
13. No, not a function
15. Yes, it is a function
17. Yes, it is a function
19. No, not a function
21.
( )
4
= −∞ ∞
= − ∞
,
[ , )
D
R
23.
( )
0
= −∞ ∞
= ∞
,
[ , )
D
R
25.
( )
( )
= −∞ ∞
= −∞ ∞
,
,
D
R
27.
( )
4
= −∞ ∞
= − ∞
,
[ , )
D
R
29.
33
12
= −
= −
[ , ]
[ , ]
D
R
31.
( )
{ }
All integers
= −∞ ∞
=
, D
R
33.
( )
1 0
= −∞ ∞
= − ∞ ⎡ ⎤
⎣ ⎦ ∪
,
[ , )
D
R
35.
( )
2 2
= −∞ ∞
= −∞ ∞
∪
,
( , ) ( , )
D
R
37.
0 4
0 2
=
=
[ , ]
[ , ]
D
R
( )
( )
31 5
3 35
13
− ∞
−∞ −
∪
∪
39. Intervals of Increasing: , ( , )
Intervals of Decreasing: , ( , )
Intervals of Constant: ( , )
41.
2 1 0
( )
5 0
x if x
f x
x if x
+ ≤ ⎧
=
⎨
− >
⎩
43.
1 2
( )
3 9 2
x if x
f x
x if x
+ ≤ ⎧
=
⎨
− + >
⎩
45.
2
3 1
( )
1
x if x
f x
x if x
≤ ⎧
=
⎨
>
⎩
Page 144
Answers
47.
3 5 2
( )
1 2
x if x
f x
if x
+ ≤ − ⎧
=
⎨
− > −
⎩
49.
2 1 2
( )
5 2
x if x
f x
if x
− ≠ ⎧
=
⎨
=
⎩
51.
2
1
( )
4 1
x if x
f x
if x
⎧ ≠
=
⎨
=
⎩
53.
6 4
1
( ) 7 4 2
2
5 2
x if x
f x x if x
x if x
+ < − ⎧
⎪
⎪
= + − ≤ ≤
⎨
⎪
− + > ⎪
⎩
55.
1 1
( ) 1 1
1 1
if x
f x x if x
if x
− < − ⎧
⎪
= − ≤ <
⎨
⎪
≥
⎩
57.
2 4 1
( ) 2 1 2
4 2
x if x
f x if x
x if x
+ ≤ − ⎧
⎪
= − < <
⎨
⎪
− + ≥
⎩
Page 145
1.7 Graphs of Functions 73
Example 1.7.4. Given the graph of y = f(x) below, answer all of the following questions.
(−2, 0) (2, 0)
(4, −3) (−4, −3)
(0, 3)
x
y
−4 −3 −2 −1 1 2 3 4
−4
−3
−2
−1
1
2
3
4
1. Find the domain of f.
2. Find the range of f.
3. Determine f(2).
4. List the x-intercepts, if any exist.
5. List the y-intercepts, if any exist.
6. Find the zeros of f.
7. Solve f(x) < 0.
8. Determine the number of solutions to the
equation f(x) = 1.
9. List the intervals on which f is increasing.
10. List the intervals on which f is decreasing.
11. List the local maximums, if any exist.
12. List the local minimums, if any exist.
13. Find the maximum, if it exists.
14. Find the minimum, if it exists.
15. Does f appear to be even, odd, or neither?
Solution.
1. To find the domain of f, we proceed as in Section 1.4. By projecting the graph to the x-axis,
we see the portion of the x-axis which corresponds to a point on the graph is everything from
−4 to 4, inclusive. Hence, the domain is [−4, 4].
2. To find the range, we project the graph to the y-axis. We see that the y values from −3 to
3, inclusive, constitute the range of f. Hence, our answer is [−3, 3].
3. Since the graph of f is the graph of the equation y = f(x), f(2) is the y-coordinate of the
point which corresponds to x = 2. Since the point (2, 0) is on the graph, we have f(2) = 0.
Page 146
Additional problems taken from Stitz and Zeager
Suggested assignment: p. 73: 1-10
1.7 Graphs of Functions 77
1.7.2 Exercises
1. Sketch the graphs of the following functions. State the domain of the function, identify any
intercepts and test for symmetry.
(a) f(x) =
x −2
3
(b) f(x) =
√
5 −x (c) f(x) =
3
√
x
(d) f(x) =
1
x
2
+ 1
2. Analytically determine if the following functions are even, odd or neither.
(a) f(x) = 7x
(b) f(x) = 7x + 2
(c) f(x) =
1
x
3
(d) f(x) = 4
(e) f(x) = 0
(f) f(x) = x
6
−x
4
+ x
2
+ 9
(g) f(x) = −x
5
−x
3
+ x
(h) f(x) = x
4
+x
3
+x
2
+x+1
(i) f(x) =
√
5 −x
(j) f(x) = x
2
−x −6
3. Given the graph of y = f(x) below, answer all of the following questions.
x
y
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
5
(a) Find the domain of f.
(b) Find the range of f.
(c) Determine f(−2).
(d) List the x-intercepts, if any exist.
(e) List the y-intercepts, if any exist.
(f) Find the zeros of f.
(g) Solve f(x) ≥ 0.
(h) Determine the number of solutions to the
equation f(x) = 2.
(i) List the intervals where f is increasing.
(j) List the intervals where f is decreasing.
(k) List the local maximums, if any exist.
(l) List the local minimums, if any exist.
(m) Find the maximum, if it exists.
(n) Find the minimum, if it exists.
(o) Is f even, odd, or neither?
Page 147
Additional problems taken from Stitz and Zeager
Suggested assignment: p. 77: 3 (a-j)
1.7 Graphs of Functions—Stitz and Zeager Book
ANSWERS p. 73:1-15
1. 4 4 [ , ] −
2. 33 [ , ] −
3. 2 0 ( ) f =
4. 20 20 ( , ),( , ) −
5. 03 ( , )
6. 2 2 , x = −
7. 4 2 2 4 [ , ] ( , ] − −
∪
8. 2 solutions
9. 4 0 [ , ) −
10. 0 4 ( , ]
11. 03 ( , )
12. none
13. 3
14. 3 −
15. yes, even
p. 77: 3 (a-j) ANSWERS
Page 148
Page 149
Cube Root Function
Page 150
Page 151
Page 152
Page 153
Page 154
Page 155
Page 156
Page 157
Page 158
Page 159
Page 160
Page 161
3.6—Additional Graphing Techniques
In problems 1-40 use the techniques of shifting, reflecting, and stretching to sketch the
graph of the following functions.
1.
2
( ) ( 1) 3 f x x 2.
2
( ) ( 1) 4 f x x
3. ( ) 1 2 f x x 4. ( ) 2 1 f x x
5.
3
( ) 3 2 f x x 6.
3
( ) 2 2 1 f x x
7. ( ) 4 4 f x x 8. ( ) 2 3 3 f x x
9.
3
( ) 2 3 f x x 10.
3
( ) 1 2 f x x
11. ( ) 2 1 1 f x x 12.
2
( ) 2 3 2 f x x
13.
2
( ) 3 f x x
14.
2
1
( )
4
f x x
15. ( ) 3 f x x 16. ( ) 1 f x x
17. ( ) 4 f x x 18. ( ) 1 f x x
19.
1
( ) 3 3
2
f x x 20.
1
( ) 2 1
2
f x x
21. ( ) 1 3 f x x
22.
1
( ) 1 4
2
f x x
23.
3
( ) 2 1 2 f x x 24. ( ) 3 f x x
25.
3
1
( ) 2 1
2
f x x
26.
3
( ) 3 f x x
27.
2
( ) 2 3 5 f x x
28.
3
1
( ) 1 4
2
f x x
29.
3
( ) 2 1 2 f x x 30.
3
( ) 2 3 f x x
31.
1
( ) 3 2
2
f x x
32. ( ) 2 1 1 f x x
33.
3 1
( ) 1 3
2
f x x 34.
1
( ) 4 2
2
f x x
35.
3
( ) 2 1 f x x
36.
3 1
( ) 4 1
2
f x x
37.
3
( ) 4 1 f x x 38.
3
( ) 3 1 f x x
39. ( ) 2 3 1 f x x
40.
3
( ) 5 3 f x x
Page 162
Basic Functions
Page 163
Page 164
Page 165
Page 166
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
Section 3.6-Graphing Techniques
1. 3. 5.
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-2 -1 1 2 3 4 5 6
-5
-4
-3
-2
-1
1
2
3
4
5
7. 9. 11.
13. 15. 17.
19. 21. 23.
-1 1 2 3 4 5 6 7
-2
-1
1
2
3
4
5
6
7
8
-1 1 2 3 4 5 6 7
-2
-1
1
2
3
4
5
6
7
8
-6 -5 -4 -3 -2 -1 1 2
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
4
5
6
7
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
Page 167
Answers
25. 27. 29.
31. 33. 35.
37. 39. 41i.
-8-7-6-5-4-3-2-1 1 2
-4
-3
-2
-1
1
2
3
4
-3 -2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
41ii. 41iii. 41iv.
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3
-5
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3
-5
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-8
-7
-6
-5
-4
-3
-2
-1
1
-2 -1 1 2 3 4 5 6
-5
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3
-2
-1
1
2
3
4
5
6
7
-6 -5 -4 -3 -2 -1 1 2
-4
-3
-2
-1
1
2
3
4
Page 168
41v. 41vi. 41vii.
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6-5-4-3-2-1 1 2 3 4 5 6
-5
-4
-3
-2
-1
1
2
3
4
5
41viii. 43i. 43ii.
-9-8-7-6-5-4-3-2-1 1 2 3
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
-4-3-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
5
6
-4-3-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
5
6
43iii. 43iv. 43v.
-4-3-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
5
6
-4-3-2-1 1 2 3 4 5 6 7 8
-6
-5
-4
-3
-2
-1
1
2
3
4
-4-3-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
5
6
Page 169
43vi. 43vii. 43viii.
-6-5-4-3-2-1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
5
6
-6-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
-4-3-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
5
6
45. even
47. neither
49. odd
51. neither
53. odd
55. even
57. even
61. neither
63. even
65. odd
67. 69. 71.
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
-2
-1
1
2
73. 75. 77.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Page 170
104 Relations and Functions
1.8.1 Exercises
1. The complete graph of y = f(x) is given below. Use it to graph the following functions.
x
y
(−2, 0)
(0, 4)
(2, 0)
(4, −2)
−4 −3 −1 1 3 4
−4
−3
−2
−1
1
2
3
4
The graph of y = f(x)
(a) y = f(x) −1
(b) y = f(x + 1)
(c) y =
1
2
f(x)
(d) y = f(2x)
(e) y = −f(x)
(f) y = f(−x)
(g) y = f(x + 1) −1
(h) y = 1 −f(x)
(i) y =
1
2
f(x + 1) −1
2. The complete graph of y = S(x) is given below. Use it to graph the following functions.
x
y
(−2, 0)
(−1, −3)
(0, 0)
(1, 3)
(2, 0)
−2 −1 1
−3
−2
−1
1
2
3
The graph of y = S(x)
(a) y = S(x + 1)
(b) y = S(−x + 1)
(c) y =
1
2
S(−x + 1)
(d) y =
1
2
S(−x + 1) + 1
Page 171
Taken from Stitz and Zeager
1.8 Transformations 105
3. The complete graph of y = f(x) is given below. Use it to graph the following functions.
(−3, 0)
(0, 3)
(3, 0)
x
y
−3 −2 −1 1 2 3
−1
1
2
3
(a) g(x) = f(x) + 3
(b) h(x) = f(x) −
1
2
(c) j(x) = f
x −
2
3
(d) a(x) = f(x + 4)
(e) b(x) = f(x + 1) −1
(f) c(x) =
3
5
f(x)
(g) d(x) = −2f(x)
(h) k(x) = f
2
3
x
(i) m(x) = −
1
4
f(3x)
(j) n(x) = 4f(x −3) −6
(k) p(x) = 4 + f(1 −2x)
(l) q(x) = −
1
2
f
x+4
2
−3
4. The graph of y = f(x) =
3
√
x is given below on the left and the graph of y = g(x) is given
on the right. Find a formula for g based on transformations of the graph of f. Check your
answer by confirming that the points shown on the graph of g satisfy the equation y = g(x).
x
y
−11−10−9−8−7−6−5−4−3−2−1 1 2 3 4 5 6 7 8
−5
−4
−3
−2
−1
1
2
3
4
5
y =
3
√
x
x
y
−11−10−9−8−7−6−5−4−3−2−1 1 2 3 4 5 6 7 8
−5
−4
−3
−2
−1
1
2
3
4
5
y = g(x)
5. For many common functions, the properties of algebra make a horizontal scaling the same
as a vertical scaling by (possibly) a different factor. For example, we stated earlier that
√
9x = 3
√
x. With the help of your classmates, find the equivalent vertical scaling produced
by the horizontal scalings y = (2x)
3
, y = |5x|, y =
3
√
27x and y =
1
2
x
2
. What about
y = (−2x)
3
, y = | −5x|, y =
3
√
−27x and y =
−
1
2
x
2
?
Page 172
1.8 Transformations 107
1.8.2 Answers
1. (a) y = f(x) −1
x
y
(−2, −1)
(0, 3)
(2, −1)
(4, −3)
−4 −3 −1 −2 1 2 3 4
−4
−3
−2
−1
1
2
3
4
(b) y = f(x + 1)
x
y
(−3, 0)
(−1, 4)
(1, 0)
(3, −2)
−4 −3 −1 −2 1 2 3 4
−4
−3
−2
−1
1
2
3
4
(c) y =
1
2
f(x)
x
y
(−2, 0)
(0, 2)
(2, 0)
(4, −1)
−4 −3 −1 1 3 4
−4
−3
−2
−1
1
2
3
4
(d) y = f(2x)
x
y
(−1, 0)
(0, 4)
(1, 0)
(2, −2)
−4 −3 −2 2 3 4
−4
−3
−2
1
2
3
4
(e) y = −f(x)
x
y
(−2, 0)
(0, −4)
(2, 0)
(4, 2)
−4 −3 −1 −2 1 2 3 4
−4
−3
−2
−1
1
2
3
4
(f) y = f(−x)
x
y
(2, 0)
(0, 4)
(−2, 0)
(−4, −2)
−4 −3 −1 1 3 4
−4
−3
−2
−1
1
2
3
4
Page 173
108 Relations and Functions
(g) y = f(x + 1) −1
x
y
(−3, −1)
(−1, 3)
(1, −1)
(3, −3)
−4 −3 −1 −2 1 2 3 4
−4
−3
−2
−1
1
2
3
4
(h) y = 1 −f(x)
x
y
(−2, 1)
(0, −3)
(2, 1)
(4, 3)
−4 −3 −1 −2 1 2 3 4
−4
−3
−2
−1
1
2
3
4
(i) y =
1
2
f(x + 1) −1
x
y
(−3, −1)
(−1, 1)
(1, −1)
(3, −2)
−4 −3 −1 −2 1 2 3 4
−4
−3
−2
−1
1
2
3
4
2. (a) y = S(x + 1)
x
y
(−3, 0)
(−2, −3)
(−1, 0)
(0, 3)
(1, 0)
−3 −2 −1
−3
−2
−1
1
2
3
(b) y = S(−x + 1)
x
y
(3, 0)
(2, −3)
(1, 0)
(0, 3)
(−1, 0)
1 2 3
−3
−2
−1
1
2
3
Page 174
1.8 Transformations 109
(c) y =
1
2
S(−x + 1)
x
y
(3, 0)
2, −
3
2
(1, 0)
0,
3
2
(−1, 0)
1 2 3
−2
−1
1
2
(d) y =
1
2
S(−x + 1) + 1
x
y
(3, 1)
2, −
1
2
(1, 1)
0,
5
2
(−1, 1)
−1 1 3
−1
1
2
3
3. (a) g(x) = f(x) + 3
(−3, 3)
(0, 6)
(3, 3)
x
y
−3 −2 −1 1 2 3
−1
1
2
3
4
5
6
(b) h(x) = f(x) −
1
2
−3, −
1
2
0,
5
2
3, −
1
2
x
y
−3 −2 −1 1 2 3
−1
1
2
3
(c) j(x) = f
x −
2
3
−
7
3
, 0
2
3
, 3
11
3
, 0
x
y
−3 −2 −1 1 2 3
−1
1
2
3
(d) a(x) = f(x + 4)
(−7, 0)
(−4, 3)
(−1, 0)
x
y
−7 −6 −5 −4 −3 −2 −1
1
2
3
(e) b(x) = f(x + 1) −1
(−4, −1)
(−1, 2)
(2, −1)
x
y
−4 −3 −2 −1 1 2
−1
1
2
(f) c(x) =
3
5
f(x)
(−3, 0)
0,
9
5
(3, 0)
x
y
−3 −2 −1 1 2 3
−1
1
2
Page 175
110 Relations and Functions
(g) d(x) = −2f(x)
(−3, 0)
(0, −6)
(3, 0)
x
y
−3 −2 −1 1 2 3
−6
−5
−4
−3
−2
−1
(h) k(x) = f
2
3
x
−
9
2
, 0
(0, 3)
9
2
, 0
x
y
−4 −3 −2 −1 1 2 3 4
−1
1
2
3
(i) m(x) = −
1
4
f(3x)
(−1, 0)
0, −
3
4
(1, 0)
x
y
−1 1
−1
(j) n(x) = 4f(x −3) −6
(0, −6)
(3, 6)
(6, −6)
x
y
1 2 3 4 5 6
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
(k) p(x) = 4 + f(1 −2x) = f(−2x + 1) + 4
(−1, 4)
1
2
, 7
(2, 4)
x
y
−1 1 2
−1
1
2
3
4
5
6
7
(l) q(x) = −
1
2
f
x+4
2
−3 = −
1
2
f
1
2
x + 2
−3
(−10, −3)
−4, −
9
2
(2, −3)
x
y
−10−9−8−7−6−5−4−3−2−1 1 2
−4
−3
−2
−1
4. g(x) = −2
3
√
x + 3 −1 or g(x) = 2
3
√
−x −3 −1
Page 176
Piecewise Functions
Graph the following:
1.
≠
=
=
2 0
( )
1 0
x if x
f x
if x
2.
3 0
( )
4 0
x if x
f x
if x
≥
=
− <
3.
2 3 1
( )
3 2 1
x if x
f x
x if x
− + <
=
− ≥
4.
2
1 0
( )
0
x if x
f x
x if x
+ <
=
≥
5.
3
2 0
( ) 3 0
0
x if x
f x if x
x if x
− ≤ <
= − =
>
6.
3 3 0
( ) 2 0
0
x if x
f x if x
x if x
+ − ≤ <
= =
>
7.
3
3
1 0
( )
0
x if x
f x
x if x
+ >
=
− ≤
8. ( )
0
2 1 0
x if x
f x
x if x
<
=
+ ≥
9. ( )
4 2
2 2 2
− <
=
− ≥
x if x
f x
x if x
10. ( )
2
1 1
1 1
− + ≤
=
− >
x if x
f x
x if x
11. ( )
1 -1
0 1 1
1 1
− − <
= − ≤ ≤
− >
x if x
f x if x
x if x
12. ( )
1 0
1 0
− ≤
=
− >
x if x
f x
if x
13. ( )
1 3
2 8 3
− ≤
=
− + >
x if x
f x
x if x
14. ( )
1
0 1
2 1
<
= =
− + >
x if x
f x if x
x if x
15. ( )
2
2 4 1
4 1
1 1
− + <
= =
+ >
x if x
f x if x
x if x
16. ( )
0
1 0
x if x
f x
if x
≠
=
=
17. ( )
1 1
2 1
− ≠
=
=
x if x
f x
if x
Page 177
Piecewise Functions
1.
2 0
( )
1 0
x if x
f x
if x
≠ ⎧
=
⎨
=
⎩
2.
3 0
( )
4 0
x if x
f x
if x
≥ ⎧
=
⎨
− <
⎩
3.
2 3 1
( )
3 2 1
x if x
f x
x if x
− + < ⎧
=
⎨
− ≥
⎩
4.
2
1 0
( )
0
x if x
f x
x if x
+ < ⎧
=
⎨
≥
⎩
5.
⎧ − ≤ <
⎪
= − =
⎨
⎪
>
⎩
3
2 0
( ) 3 0
0
x if x
f x if x
x if x
6.
3 3 0
( ) 2 0
0
x if x
f x if x
x if x
⎧
+ − ≤ <
⎪
= =
⎨
⎪
>
⎩
7.
⎧
+ >
⎪
=
⎨
− ≤ ⎪
⎩
3
3
1 0
( )
0
x if x
f x
x if x
8. ( )
0
2 1 0
x if x
f x
x if x
< ⎧
=
⎨
+ ≥
⎩
9. ( )
4 2
2 2 2
− < ⎧
=
⎨
− ≥
⎩
x if x
f x
x if x
Page 178
Answers
10. ( )
2
1 1
1 1
÷ + s ¦
=
´
÷ >
¹
x if x
f x
x if x
11. ( )
1 -1
0 1 1
1 1
x if x
f x if x
x if x
÷ ÷ < ¦
¦
= ÷ s s
´
¦
÷ >
¹
12. ( )
1 0
1 0
x if x
f x
if x
÷ s ¦
=
´
÷ >
¹
13. ( )
1 3
2 8 3
÷ s ¦
=
´
÷ + >
¹
x if x
f x
x if x
14. ( )
1
0 1
2 1
x if x
f x if x
x if x
< ¦
¦
= =
´
¦
÷ + >
¹
15. ( )
2
2 4 1
4 1
1 1
x if x
f x if x
x if x
÷ + < ¦
¦
= =
´
¦
+ >
¹
16. ( )
0
1 0
x if x
f x
if x
¦ =
=
´
=
¹
17. ( )
1 1
2 1
¦ ÷ =
=
´
=
¹
x if x
f x
if x
Page 179
Page 180
Page 181
Page 182
Page 183
Page 184
Page 185
Page 186
Page 187
Page 188
Page 189
Page 190
Page 191
Page 192
Page 193
Page 194
Page 195
Page 196
Page 197
Page 198
Page 199
y =0
y =-3
Section 3.7-Linear Functions
1. 3. 5.
7. 9.
11.
5
4
m
−
=
13.
2
9
m=
15. m undefined =
17. 2 m=
19. 1 m= −
21. 23. 25.
y =
⎝
⎜
⎛ 1
2
⎠
⎟
⎞
x +3
y =3x
x =3
f(x) =2x – 4
y =
⎝
⎜
⎛ 1
3
⎠
⎟
⎞
x –
14
3 y =–2x +7
Page 200
Answers
27.
29.
1 14
3 3
= − y x
31. 2 7 = − + y x
33. 3 = − y
35. 4 = − x
37.
5 29
4 4
−
= + y x
39.
2 1
9 9
= − y x
41. 4 = x
43. 2 = y
45.
2
2
5
= − y x
47.
2
7
m
b
= −
=
49.
3
5
2
m
b
=
= −
51.
0
4
m
b
=
=
53.
m undefined
b none
=
=
55.
1
2
3
m
b
=
=
57. 4 1 = + y x
59.
3 9
5 5
−
= + y x
61.
4 8
3 3
= + y x
63. 3 = − x
65. 5 = y
67.
1 21
4 4
−
= + y x
69.
5 19
3 3
= + y x
71.
3 31
4 4
−
= − y x
73. 4 = y
75. 3 = x
x =–4
y =–2x +7
y =
⎝
⎜
⎛ 3
5
⎠
⎟
⎞
x – 2
y =4
x =–2
y =
⎝
⎜
⎛ 1
2
⎠
⎟
⎞
x +3
Page 201
Page 202
Page 203
Page 204
Page 205
Page 206
Page 207
Page 208
Page 209
Page 210
Page 211
Page 212
Page 213
Section 3.8—Circles
1. 3. 5.
7. 9. 11.
13. 15. 17.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
2
+y
2
=1
-5-4-3-2-1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
(x – 1)
2
+(y +1)
2
=16
-1 1 2 3 4 5 6 7 8 910
-4
-3
-2
-1
1
2
3
4
5
6
(x – 5)
2
+(y – 2)
2
=9
-5-4-3-2-1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
7
8
(x – 1)
2
+(y – 3)
2
=20
-7-6-5 -4-3-2 -1 1 2
-7
-6
-5
-4
-3
-2
-1
1
2 (x +3)
2
+(y +4)
2
=5
-3 -2 -1 1 2 3
-2
-1
1
2
3
4
⎝
⎜
⎛
x +
1
2
⎠
⎟
⎞
2
+
⎝
⎜
⎛
y –
5
3
⎠
⎟
⎞
2
=
9
4
-10 -8 -6 -4 -2 2 4 6
-4
-2
2
4
6
8
10
(x +3)
2
+(y – 2)
2
=25
-7-6-5 -4-3-2 -1 1 2
-7
-6
-5
-4
-3
-2
-1
1
2
(x +2)
2
+(y +5)
2
=5
-10 -8 -6 -4 -2 2 4 6
-2
2
4
6
8
10
12
(x +2)
2
+(y – 5)
2
=41
Page 214
19. 21. 23.
25. 27. 29. No graph
31. ( 5,3) −
33. 35.
1 5
,
3 3
− ⎛ ⎞
⎜ ⎟
⎝ ⎠
37.
-4 -3 -2 -1 1 2 3 4
-1
1
2
3
4
5
6
7
8 r =
1
2
C=(1/2,5/2)
-8 -7 -6 -5 -4 -3 -2 -1 1 2
-4
-3
-2
-1
1
2
3
4
5
6
r =
70
2
C=(-3,1)
39. No graph
41.
2 2
( 2) ( 5) 25 x y + + − =
43.
2 2
( 6) ( 6) 36 x y − + + =
45.
3 25
4 4
y x
−
= +
-2 2 4 6
-8
-7
-6
-5
-4
-3
-2
-1
1
r =2
C=( 5 , -3 )
-6 -4 -2 2
-6
-5
-4
-3
-2
-1
1
r = 7
C=( -2 , -3 )
-6-5-4-3-2-1 1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
11
12
r = 13 C=( 0 , 7 )
-4 -2 2 4
-6
-5
-4
-3
-2
-1
1
2
r = 6
C=( 0 , -2 )
-4 -2 2 4 6 8 10
-4
-2
2
4
6
8
10
r =5
C=( 3 , 2 )
Page 215
Circles
Complete the square and write the equation in standard form. Then
give the center and radius of each circle.
1.
2 2
41
5 8 0
4
x y x y + + − + =
2.
2 2
3 295
2 0
2 16
x y x y + − + − =
3.
2 2
45
3 0
2
x y x y + − + − =
4.
2 2
21
6 7 0
4
x y x y + − + + =
5.
2 2
79
4 0
4
x y x y + + − − =
6.
2 2
27
9 3 0
2
x y x y + − − + =
7.
2 2
1 2 2567
0
2 3 144
x y x y + + + − =
8.
2 2
10 35
0
3 36
x y x y + + + − − =
Find an equation of the circle that satisfies the given conditions.
9. Center ( ) 8, 3 − ; tangent to the x-axis.
10. Center ( ) 4,5 − ; tangent to the x-axis.
11. Center ( ) 8, 3 − ; tangent to the y-axis.
12. Center ( ) 4,5 − ; tangent to the y-axis.
13. Center at the origin; passes through ( ) 5, 3 −
14. Center at the origin; passes through ( ) 2,7 −
15. Endpoints of the diameter are P( ) 1,1 − and Q( ) 5,5
16. Endpoints of the diameter are P( ) 1,3 − and Q( ) 7, 5 −
17. Endpoints of the diameter are P( ) 3,4 and Q( ) 5,1
18. Endpoints of the diameter are P( ) 3, 8 − − and Q( ) 6,6
Page 216
Circles-Answers
1. ( )
2
2 5
4 12
2
x y
+ + − =
; Center =
5
,4
2
−
; r=2 3
2. ( )
2
2 3
1 20
4
x y
− + + =
; Center =
3
, 1
4
−
; r=2 5
3.
2 2
1 3
25
2 2
x y
− + + =
; Center =
1 3
,
2 2
−
; r=5
4. ( )
2
2 7
3 16
2
x y
− + + =
; Center =
7
3,
2
−
; r=4
5. ( )
2
2 1
2 24
2
x y
+ + − =
; Center =
1
2,
2
−
; r=2 6
6.
2 2
9 3
9
2 2
x y
− + − =
; Center =
9 3
,
2 2
; r=3
7.
2 2
1 1
18
4 3
x y
+ + + =
; Center =
1 1
,
4 3
− −
; r=3 2
8.
2 2
5 1
4
3 2
x y
+ + − =
; Center =
5 1
,
3 2
−
; r=2
9. ( ) ( )
2 2
8 3 9 x y − + + =
10. ( ) ( ) + + − =
2 2
4 5 25 x y
11. ( ) ( )
2 2
8 3 64 x y − + + =
12. ( ) ( )
2 2
4 5 16 x y + + − =
13.
2 2
34 x y + =
14.
2 2
53 x y + =
15. ( ) ( )
2 2
2 3 13 x y − + − =
16. ( ) ( )
2 2
3 1 32 x y − + + =
17. ( )
2
2 3 13
4
2 4
x y
− + − =
18. ( )
− + + =
2
2 3 277
1
2 4
x y
Page 217
Page 218
Page 219
Page 220
Page 221
Page 222
Page 223
Page 224
Page 225
Page 226
Page 227
Page 228
Page 229
Page 230
Page 231
Page 232
In problems 61-66, use the given functions f and g to find the indicated function values.
61. ( )( 3) f g − D 62. ( )(1) g f D
63. ( )(3) f g D 64. ( )(7) f g D
65. ( )( 5) g f − D 66. ( )(3) g f D
Additional problems :
A. ( )(0) f g D B. ( )(9) g f D
C. ( )( 10) f g − D D. ( )( 1) f g − D
E. ( )(3) g f D F. ( )(6) g f D
x
y
g(x)
x
y
f(x)
Page 233
Section 3.9-Operations on Functions
1.
2
3 1
5 2 4 6 7 3
2 3
( )( ) ; ( )( ) ; ( )( ) ; ( )
f x
f g x x f g x x fg x x x x
g x
+
+ = − − = + = − − =
−
3.
2 2 3 2
2
2 4
2 1 2 7 2 4 6 12
3
( )( ) ; ( )( ) ; ( )( ) ; ( )
f x
f g x x x f g x x x fg x x x x x
g x
−
+ = + − − = − + − = − + − =
+
5.
2
2 2 3 2
5 2
6 4 4 8 28 12
6
( )( ) ; ( )( ) ; ( )( ) ; ( )
f x x
f g x x x f g x x x fg x x x x x
g x
+ +
+ = + − − = + + = − − − =
−
7.
( )( ) ( )( ) ( )( ) ( )
3 1 7 2 2 4
3 2 3 2 3 2 3
+ + +
+ = − = = =
− + − + − + −
( )( ) ; ( )( ) ; ( )( ) ; ( )
x x f x
f g x f g x fg x x
g x x x x x x x
9.
2
2 2 3 2
20
2 15 25 6 15 100 4
5
+ −
+ = + − − = − = + − − = = −
+
( )( ) ; ( )( ) ; ( )( ) ; ( )
f x x
f g x x x f g x x fg x x x x x x
g x
11.
2 2 2
2
4 4 4
4
( )( ) ; ( )( ) ; ( )( ) ( ); ( )
( )
f x
f g x x x f g x x x fg x x x x
g x
+ = + − − = − + = − =
−
13.
2 2
18 48 39 6 17 ( )( ) ; ( )( ) f g x x x g f x x = − + = +
15.
2 2
10 24 4 4 ( )( ) ; ( )( ) f g x x x g f x x x = − + = + −
17.
2
2 2 ( )( ) ; ( )( ) f g x x g f x x = − = −
19.
3 1 1
3
( )( ) ; ( )( )
x
f g x g f x
x x
+
= =
+
21.
3 3 4 8
2 2 1
( )( ) ; ( )( )
x x
f g x g f x
x x
+ −
= =
− + +
23.
1 1
3 3
( )( ) ; ( )( ) f g x g f x
x x
= =
− −
25. 9 4 x +
27. 32 −
29.
( )
, −∞ ∞
31. 2 −
33.
2
36 42 16 x x − +
35. 76
37. 3 −
47.
2
2 37 ( ( ))( ) (( ) ))( ) f g h x f g h x x = = −
49. 1
51. 1
53. 8 −
55. 2
57. 3
59. 4
61. 1 −
63. 1 −
65. 1
Page 234
Answers
60 Relations and Functions
1.6.1 Exercises
1. Let f(x) =
√
x, g(x) = x + 10 and h(x) =
1
x
.
(a) Compute the following function values.
i. (f + g)(4) ii. (g −h)(7) iii. (fh)(25)
iv.
h
g
(3)
(b) Find the domain of the following functions then simplify their expressions.
i. (f + g)(x)
ii. (g −h)(x)
iii. (fh)(x)
iv.
h
g
(x)
v.
g
h
(x)
vi. (h −f)(x)
2. Let f(x) = 3
√
x −1, g(x) = 2x
2
−3x −2 and h(x) =
3
2 −x
.
(a) Compute the following function values.
i. (f + g)(4) ii. (g −h)(1) iii. (fh)(0)
iv.
h
g
(−1)
(b) Find the domain of the following functions then simplify their expressions.
i. (f −g)(x) ii. (gh)(x)
iii.
f
g
(x) iv.
f
h
(x)
3. Let f(x) =
√
6x −2, g(x) = x
2
−36, and h(x) =
1
x −4
.
(a) Compute the following function values.
i. (f + g)(3)
ii. (g −h)(8)
iii.
f
g
(4)
iv. (fh)(8)
v. (g + h)(−4)
vi.
h
g
(−12)
(b) Find the domain of the following functions and simplify their expressions.
i. (f + g)(x)
ii. (g −h)(x)
iii.
f
g
(x)
iv. (fh)(x)
v. (g + h)(x)
vi.
h
g
(x)
Page 235
Taken from Stitz and Zeager
62 Relations and Functions
1.6.2 Answers
1. (a) i. (f + g)(4) = 16
ii. (g−h)(7) =
118
7
iii. (fh)(25) =
1
5
iv.
h
g
(3) =
1
39
(b) i. (f + g)(x) =
√
x + x + 10
Domain: [0, ∞)
ii. (g −h)(x) = x + 10 −
1
x
Domain: (−∞, 0) ∪ (0, ∞)
iii. (fh)(x) =
1
√
x
Domain: (0, ∞)
iv.
h
g
(x) =
1
x(x + 10)
Domain: (−∞, −10)∪(−10, 0)∪(0, ∞)
v.
g
h
(x) = x(x + 10)
Domain: (−∞, 0) ∪ (0, ∞)
vi. (h −f)(x) =
1
x
−
√
x
Domain: (0, ∞)
2. (a) i. (f + g)(4) = 23 ii. (g −h)(1) = −6
iii. (fh)(0) = −
3
2
iv.
h
g
(−1) =
1
3
(b) i. (f −g)(x) = −2x
2
+ 3x + 3
√
x + 1
Domain: [0, ∞)
ii. (gh)(x) = −6x −3
Domain: (−∞, 2) ∪ (2, ∞)
iii.
f
g
(x) =
3
√
x −1
2x
2
−3x −2
Domain: [0, 2) ∪ (2, ∞)
iv.
f
h
(x) = −x
√
x +
1
3
x + 2
√
x −
2
3
Domain: [0, 2) ∪ (2, ∞)
3. (a) i. (f + g)(3) = −23
ii. (g −h)(8) =
111
4
iii.
f
g
(4) = −
√
22
20
iv. (fh)(8) =
√
46
4
v. (g + h)(−4) = −
161
8
vi.
h
g
(−12) = −
1
1728
(b) i. (f + g)(x) = x
2
−36 +
√
6x −2
Domain:
1
3
, ∞
ii. (g −h)(x) = x
2
−36 −
1
x −4
Domain: (−∞, 4) ∪ (4, ∞)
iii.
f
g
(x) =
√
6x −2
x
2
−36
Domain:
1
3
, 6
∪ (6, ∞)
iv. (fh)(x) =
√
6x −2
x −4
Domain:
1
3
, 4
∪ (4, ∞)
v. (g + h)(x) = x
2
−36 +
1
x −4
Domain: (−∞, 4) ∪ (4, ∞)
vi.
h
g
(x) =
1
(x −4) (x
2
−36)
Domain:
(−∞, −6) ∪ (−6, 4) ∪ (4, 6) ∪ (6, ∞)
Page 236
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Page 250
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Page 253
Section 3.10—Inverse Functions
1. one-to-one
3. not one-to-one
5. one-to-one
7. one-to-one
9. not one-to-one
11. one-to-one
13. not one-to-one
15. not one-to-one
17. not one-to-one
19. inverses
21. not inverses
23. not inverses
25. not inverses
27. inverses
29. inverses
31. not inverses
33. 35. 37.
39. 41. 43.
45. 47.
49.
1
3 3
( )
2 4
f x x
−
= +
51.
1
3
4
( )
5
x
f x
−
+
=
53. not one-to-one
55.
1 2
( ) 4, 0 f x x x
−
= − ≥
57.
1
( ) 2 f x x
−
= +
59.
1
( ) 1 f x x
−
= − −
61. not one-to-one
63. not one-to-one
65.
( )
3
1
( ) 1 f x x
−
= −
67.
1
2 1
( )
x
f x
x
−
+
=
69.
( ) 4000
900
C x
x
−
=
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
f
-1
(x) =x +4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
f
-1
(x) =
x – 5
2
-2-1 1 2 3 4 5 6 7 8
-2
-1
1
2
3
4
5
6
7
8
f
-1
(x) =x
2
+4;x ≥ 0
-4 -3 -2 -1 1 2 3 4
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4 f
-1
(x) = x – 2
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
f
-1
(x) =
3
x – 2
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-5-4-3-2-1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4
Page 254
Answers
Page 255
Page 256
Page 257
Page 258
Page 259
Page 260
Page 261
Page 262
Page 263
Page 264
Page 265
Page 266
Page 267
Section 4.1-Quadratic Functions
1. 2
3. 38
5. 29
7. 51
11. yes
13. no
15. no
17.
3 13
2 2
= , x
19. 3 = − x
21. 1 3 = ± x
23. Axis: 0 x = ;
Range: ( ,0] −∞
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
Vertex
( 0 , 0 )
25. Axis: 0 x = ;
Range: [ 3, ) − ∞
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
Vertex
( 0 , -3 )
27. Axis: 2 x = ;
Range: ( ,0] −∞
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
Vertex
( 2 , 0 )
29. Axis: 2 x = − ;
Range: [2, ) ∞
-4 -3 -2 -1 1 2
-2
-1
1
2
3
4
Vertex
( -2 , 2 )
31. Axis:
1
2
x = ;
Range: ( ,1] −∞
-3 -2 -1 1 2 3
-3
-2
-1
1
2
Vertex
(1/2,1)
33. Axis: 5 x = − ;
Range: [ 2, ) − ∞
-8 -7 -6 -5 -4 -3 -2 -1 1 2
-3
-2
-1
1
2
3
Vertex
( -5 , -2 )
35. x-int(s): 1 6 x = − ±
2
( ) ( 1) 6 f x x = + −
-4 -3 -2 -1 1 2 3 4
-8
-7
-6
-5
-4
-3
-2
-1
1
2
Vertex
( -1 , -6 )
37. x-int(s): none
2
( ) ( 1) 7 = − + − f x x
-4 -3 -2 -1 1 2 3 4
-12
-10
-8
-6
-4
-2
2
Vertex
( -1 , -7 )
39. x-int(s): 0,2 x =
2
( ) 4( 1) 4 = − − + f x x
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
Vertex
( 1 , 4 )
Page 268
Answers
41. x-int(s): 4 x
2
( ) ( 4) f x x
-2 -1 1 2 3 4 5 6
-2
-1
1
2
3
Vertex
( 4 , 0 )
43. x-int(s): none
2
( ) ( 4) 1 f x x
-6 -5 -4 -3 -2 -1 1 2
-2
-1
1
2
3
Vertex
( -4 , 1 )
45. x-int(s): none
2
1 3
( ) ( )
2 4
f x x
-3 -2 -1 1 2 3 4
-2
-1
1
2
3
Vertex
(1/2,3/4)
47. x-int(s):
3 19
2
x
2
3 19
( ) ( )
2 2
f x x
-6 -5 -4 -3 -2 -1 1 2 3
-10
-8
-6
-4
-2
2
Vertex
(-3/2,-19/2)
49. x-int(s):
2
4,
3
x
2
5 49
( ) ( )
3 3
f x x
-4 -3 -2 -1 1 2
-4
-2
2
4
6
8
10
12
14
16
Vertex
(-5/3,49/3)
51. x-int(s): 0,9 x
2
9 27
( ) ( )
2 2
f x x
-2 -1 1 2 3 4 5 6 7 8 9 10
-4
-2
2
4
6
8
10
12
14
16
Vertex
(9/2,27/2)
53. x-int(s): 5,4 x ,
Minimum value =
81
4
55. x-int(s):
3
2
x ,
Minimum value = 9
57. x-int(s):
3
2
x ,
Minimum value = 0
59. x-int(s): none,
Maximum value = 2
61. x-int(s): 0,9 x ,
Maximum value =
405
4
63.
2 1
( ) 1 2
4
f x x
65. 8 c
Page 269
Quadratic Functions-Worksheet
Find the vertex and “a” and then use to sketch the graph of each function.
Find the intercepts, axis of symmetry, and range of each function.
Remember the domain is ( , )
1.
2
( ) 4 1 f x x
2.
2
( ) 3 1 f x x
3.
2 1
( ) 1 2
2
f x x
4.
2 1
( ) 3 1
2
f x x
5.
2
( ) 2 3 f x x
6.
2
( ) 2 2 1 f x x
7.
2
( ) 1 4 f x x
8.
2
( ) 2 2 f x x
9.
2
( ) 2 5 f x x x
10.
2
( ) 2 8 f x x x
11.
2
( ) 4 7 f x x x
12.
2
( ) 2 3 f x x x
13.
2
( ) 6 3 f x x x
14.
2
( ) 2 4 3 f x x x
15.
2
( ) 2 2 f x x x
<<Worksheet-1>>
Quadratic Functions Worksheet--Answers
1. x-int(s): 3, 5 x
Axis: 4 x ;
Range: [ 1, )
-3 -2 -1 1 2 3 4 5 6 7 8
-3
-2
-1
1
2
3
4
5
Vertex
( 4 , -1 )
2. x-int(s): 2, 4 x
Axis: 3 x ;
Range: ( , 1]
-6 -5 -4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Vertex
( -3 , 1 )
3. x-int(s): NONE
Axis: 1 x ;
Range: [2, )
4. x-int(s): NONE
Axis: 3 x ;
Range: [1, )
5. x-int(s): 2 3 x
Axis: 2 x ;
Range: [ 3, )
-6 -5 -4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
5
6
Vertex
( -2 , -3 )
6. x-int(s):
1
2
2
x
Axis: 2 x ;
Range: [ 1, )
-5 -4 -3 -2 -1 1 2 3
-3
-2
-1
1
2
3
4
5
Vertex
( -2 , -1 )
7. x-int(s): 1, 3 x
Axis: 1 x ;
Range: ( ,4]
-4 -3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4
5
Vertex
( 1 , 4 )
8. x-int(s): 2 2 x
Axis: 2 x ; Range: ( ,2]
-6 -5 -4 -3 -2 -1 1 2 3
-4
-3
-2
-1
1
2
3
4
5
Vertex
( -2 , 2 )
9. x-int(s): 1 6 x
Axis: 1 x ;
Range: [ 6, )
2
( ) ( 1) 6 f x x
-4 -3 -2 -1 1 2 3 4
-8
-7
-6
-5
-4
-3
-2
-1
1
2
Vertex
( -1 , -6 )
<<Worksheet-1>>
Quadratic Functions Worksheet--Answers
10. x-int(s): none
2
( ) ( 1) 7 f x x
Axis: 1 x ;
Range: ( , 7]
-4 -3 -2 -1 1 2 3 4
-12
-10
-8
-6
-4
-2
2
Vertex
( -1 , -7 )
11. x-int(s): 1 6 x
2
( ) ( 2) 11 f x x
Axis: 2 x ;
Range: [ 11, )
-6 -5 -4 -3 -2 -1 1 2 3
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
Vertex
( -2 , -11 )
12. x-int(s): 1, 3 x
2
( ) ( 1) 4 f x x
Axis: 1 x ;
Range: ( ,4]
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
Vertex
( 1 , 4 )
13. x-int(s): 3 6 x
2
( ) ( 3) 6 f x x
Axis: 3 x ;
Range: [ 6, )
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3
-6
-5
-4
-3
-2
-1
1
2
3
4
Vertex
( -3 , -6 )
14. x-int(s):
2 10
2
x
2
( ) ( 1) 5 f x x
Axis: 1 x ;
Range: [ 5, )
-5 -4 -3 -2 -1 1 2 3
-6
-5
-4
-3
-2
-1
1
2
3
4
Vertex
( -1 , -5 )
15. x-int(s): NONE
2
( ) ( 1) 1 f x x
Axis: 1 x ;
Range: ( , 1]
-5 -4 -3 -2 -1 1 2 3 4 5
-7
-6
-5
-4
-3
-2
-1
1
2
3
Vertex
( 1 , -1 )
<<Worksheet-1>>
Page 270
Page 271
Page 272
Page 273
Page 274
Page 275
Page 276
Page 277
Page 278
Page 279
Page 280
Page 281
Page 282
Page 283
Page 284
Page 285
Section 4.2-Graphs of Higher Degree Polynomial Functions
1. Yes, Degree=1;
Leading coefficient=5
3. Yes, Degree=2;
Leading coefficient=1
5. No
7. Yes, Degree=3;
Leading coefficient=1
9. Yes, Degree=4;
Leading coefficient=30
11.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
13. (a) even (b) neither
(c) even (d) odd
15. 2 a
17. (a)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
17.
(b)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
(c)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
(d)
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
17.
(e)
-3 -2 -1 1 2 3
-2
-1
1
2
3
4
5
6
7
8
19.
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
7
8
21.
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
7
8
Page 286
Answers
23. 25. 27.
29. 31. 33.
35. 37. 39.
41. 43. 45.
-4 -3 -2 -1 1 2 3 4
-8
-7
-6
-5
-4
-3
-2
-1
1
2
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
7
8
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-4 -3 -2 -1 1 2 3 4
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3
-3
-2
-1
1
2
3
Page 287
47. 49. 51.
53. 55. ( ,1) (1, )
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
Page 288
Page 289
Page 290
Page 291
Page 292
Page 293
Page 294
Page 295
Page 296
Page 297
Page 298
Page 299
Section 4.3-Division of Polynomials
1.
6
2 1
5
x
x
− +
+
3. 3 6 x +
5. 9 8 x −
7.
2
2
5 4
3 1
x
x
+ +
−
9.
2
5
9 7 2
5
x x
x
− − −
−
11.
3
2 6
3
x
x
− +
+
13. 3 2 x +
15.
2
5
2 3 4
2 3
x x
x
+ + +
−
17.
3 2
5
2 3 1
2 6
x x x
x
− + + −
+
19.
2
2
2 1
2 4
3
x
x x
x x
−
− + +
+ −
21.
3 2
1 x x x + + +
23.
2
4
2 1
2 3
x
x x
+ −
− +
25.
2
2
3 5
6 2
2 4
x
x x
x
+
− + +
−
27.
4 3 2
2
2
2 2 3
3 1
x x x x
x
− + − + −
+
29.
3
1
2
x
x
− +
−
31.
17
2 4
2
x
x
− +
+
33.
2
6
2 4
3
x x
x
− − −
−
35.
2
2
2 3
2
x x
x
− − +
+
37.
2
4
3 1
4
x x
x
− − +
+
39.
2
3
4 2 4
1
2
x x
x
− − +
+
41.
3 2
2 4 6 x x x − − −
43.
3 2
5
16 12 4
3
4
x x
x
− + −
+
45.
2
5 25 x x − +
47.
5 4 3 2
2 4 8 16 32 x x x x x + + + + +
49. 3 x +
51.
3 2
12
2 2 2 2
2
x x x
x
− + − −
+
Page 300
Answers
Page 301
Page 302
Page 303
Page 304
Page 305
Page 306
Page 307
Page 308
Page 309
Page 310
Page 311
Page 312
#44.This should say 2 and -1 are zeros
Section 4.4—The Remainder and Factor Theorems
1. 34
3. 316 −
5. 0
7.
310
81
9. 11
11. 174
13. 5 2 2 +
15. 8
17. Yes, it is a zero
19. No, it is not a zero
21. No, it is not a zero
23. No, it is not a zero
25. Yes, it is a factor
27. Yes, it is a zero
29. No, it is not a factor
31. No, it is not a factor
33. No, it is not a factor
35. Yes, it is a factor
37. 2 3 1 1 − + − ( )( )( ) x x x
39.
2
5 2 3 1 + + − ( ) ( )( ) x x x
41.
3
4 3 1 + − ( )( ) x x
43.
2
1 2 1 + + − ( )( )( ) x x x
45.
3 2
2 29 30 − − + x x x
47.
3 2
1 3
3
2 2
− − − x x x or
3 2
2 6 3 − − − x x x
49.
4 3 2
14
12
3
− + + x x x x or
4 3 2
3 14 3 36 − + + x x x x
53.
27
2
−
= k
55. 2 = − k
Page 313
Answers
Page 314
Page 315
Page 316
Page 317
Page 318
Page 319
Page 320
Page 321
Page 322
Page 323
Page 324
For an alternate method for
29-48 please see page 337
Page 325
Section 4.5—Real Zeros of Polynomial Functions
1. (a)
5
1 3
2
, , x ,(b)
1
4
3
, x , (c) 0 2 , x (d) 1 3 x
21. 1 3 9 , ,
23.
1 2 5 10
1 2 5 10
3 3 3 3
, , , , , , ,
25.
1 3
1 2 3 6
2 2
, , , , ,
27.
1 1 1
1 2
8 2 4
, , ,
29. 5 2 1 ( )( )( ) x x x
31. 1 2 1 1 ( )( )( ) x x x
33.
2
2 1 ( ) ( ) x x
35.
2 2 2 1 3 1 ( )( )( ) x x x x
37.
2 2
1 1 2 3 ( )( ) ( ) x x x
39.
1
6
2
, x
41. 1 5 2 , x
43.
2
1 2 1
3
, , x
45.
3
5 2 , x
47.
1 5 1
1
2 4
, , x
51.
2
6 2 1 ( )( ) x x
53.
2
2 4 2 ( )( ) x x x
55.
2
2 1 1 3 2 ( )( ) x x x x
Page 326
Answers
Page 327
Page 328
Page 329
Page 330
Page 331
Page 332
Page 333
Page 334
f(x) = x
3
− 10x − 12
f(x) = 4x
3
− 11x − 7
f(x) = 2x
4
+ 3x
3
+ x
2
− 20x − 20
For an alternate method for
1-20, please see page 337
Page 335
Page 336
Exercises 4.5—Graphical Approach-Alternate Method
I n exer ci ses 29- 47, use t he Rat i onal Zer os Theor em, t he gi ven gr aph, and
synt het i c di vi si on t o f i nd al l zer os of each pol ynomi al f unct i on.
29.
3 2
( ) 4 7 10 f x x x x 31.
3 2
( ) 2 2 1 f x x x x
33.
3 2
( ) 3 4 f x x x 35.
4 3 2
( ) 6 25 4 4 f x x x x x
37.
5 4 3 2
( ) 4 8 7 17 3 9 f x x x x x x 39.
3 2
( ) 2 12 6 f x x x x
41.
3 2
( ) 4 8 f x x x 43.
4 3 2
( ) 3 5 7 3 2 f x x x x x
-5 -4 -3 -2 -1 1 2 3 4 5
-5 -4 -3 -2 -1 1 2 3 4 5
-5 -4 -3 -2 -1 1 2 3 4 5
-5 -4 -3 -2 -1 1 2 3 4 5
-2 -1 1 2
-5 -4 -3 -2 -1 1 2 3 4 5
-5 -4 -3 -2 -1 1 2 3 4 5
-3 -2 -1 1 2 3
Page 337
45.
4 3
( ) 2 5 10 f x x x x 47.
4 3 2
( ) 4 7 2 4 1 f x x x x x
Exercises 4.6— Graphical Approach-Alternate Method
I n exer ci ses 9- 19, use t he Rat i onal Zer os Theor em, t he gi ven gr aph, and
synt het i c di vi si on t o f i nd al l zer os of each pol ynomi al f unct i on.
9.
3
( ) 10 12 f x x x 11.
3
( ) 4 11 7 f x x x
13.
4 3 2
( ) 2 3 20 20 f x x x x x 15.
4 2
( ) 4 12 9 f x x x x
17.
4 3 2
( ) 4 12 13 12 9 f x x x x x 19.
5 4 3 2
( ) 3 3 9 4 12 f x x x x x x
-3 -2 -1 1 2 3
-3 -2 -1 1 2 3
-3 -2 -1 1 2 3 4 5
-5 -4 -3 -2 -1 1 2 3 4 5
-4 -3 -2 -1 1 2 3 4
-3 -2 -1 1 2 3 4 5
-3 -2 -1 1 2 3
-3 -2 -1 1 2 3
Page 338
Section 4.6—Complex Zeros of Pol ynomial Functions
1.
9
5
= x
3.
1 17
4
±
= x
5. 1 3 = ± , x i
7. 0 3 = ± , x i
9. 21 7 = − ± , x
11.
1 2 2
1
2
±
= − , x
13.
5 55
12
4
− ±
= − , ,
i
x
15. 311 2 = − ± , , x i
17.
3
2
= ± , x i (multiplicity of 2)
19. 2 3 = ± ± , , x i
21.
( )
2
9 1 1 3 3 + − = − − + ( ) ( )( )( ) x x x x i x i
23.
( )
2
5 55 5 55
2 2 5 10 2
4 4 4 4
− + + = − + − + +
( ) ( )
i i
x x x x x x x x
25.
( )
2 2 2
1 2 3 2 3 − + − = − − + − ( ) ( ) ( )( ) x x x x i x i
27.
( )( )
2 2
5 1 5 5 − + = − + − + ( )( )( )( ) x x x x x i x i
29.
( )( )
2 2
2 2 4 1 1 2 2 − + + = − + − − − + ( )( )( )( ) x x x x i x i x i x i
31.
( )( )
2
10 26 1 1 5 5 − + − = − − + − − ( )( )( ) x x x x x i x i
33.
3 2
16 16 + + + x x x
35.
4 3 2
10 38 64 40 − + − + x x x x
37.
4 3 2
14 98 406 841 − + − + x x x x
39.
4 2
8 16 + + x x
Page 339
Answers
Page 340
Page 341
Page 342
Page 343
Page 344
Page 345
Page 346
Page 347
Page 348
Page 349
Page 350
Page 351
#15.Correction
Page 352
Extra Problems:
Page 353
Section 4.8—Rational Functions
1. Rational
3. Not
5. Not
7. Rational
9.
1
3
= x
11.
1
2
= ± x
13.
( )
÷· · ,
15. 12 = , x
17. Vertical: 3 = x ; Horizontal: 5 = y
19. Vertical: 2 = ÷ x ; Horizontal: none
21. Vertical: none; Horizontal: 0 = y
23. Vertical:
2
1
3
= ÷ , x ; Horizontal:
1
3
= y
25. Vertical: 4 = ÷ x ; Horizontal:
0 = y ; Intercepts:
1
0
2
| |
|
\ .
,
27. Vertical: 1 = x ; Horizontal:
3 = y ; Intercepts:
( )
00 ,
29. Vertical: 2 = ± x ;
Horizontal: 0 = y ;
Intercepts:
( )
0 0 ,
31. Vertical: none; Horizontal:
0 = y ; Intercepts:
( )
0 0 ,
33. Vertical: 3 1 = ÷ , x ;
Horizontal: 2 = y ; Intercepts:
( )
0 0 ,
35. Vertical: 0 = x ;
Horizontal: 1 = y ;
Intercepts:
( )
2 0 ± ,
-7 -6 -5 -4 -3 -2 -1 1 2 3
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-7 -6 -5 -4 -3 -2 -1 1 2 3
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
Page 354
Answers
37. Vertical: 1 x ; Horizontal:
0 y ; Intercepts:
0 0 ,
39. Vertical: none; Horizontal:
1 y ; Intercepts:
0 0 ,
47. 49. 51.
53. 55.
-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
-4 -3 -2 -1 1 2 3 4
y
-4
-3
-2
-1
1
2
3
4
x
-6 -5 -4 -3 -2 -1 1 2 3 4
y
-4
-3
-2
-1
1
2
3
4
Page 355
Answers
EXTRA PROBLEMS
2
2
( )
2
f x
x x
y-int: (0, 1) x-int(s): none ;
Vertical: 21 , x ; Horizontal: 0 y
2
2
3 4
( )
2 5
x x
f x
x x
y-int: none x-int(s):
4
,1
3
;
Vertical:
5
0
2
, x ; Horizontal:
3
2
y
2
2
( )
6
x
f x
x x
y-int:
1
3
x-int(s): 2 ;
Vertical: 32 , x ; Horizontal: 0 y
2
2
12
( )
4
x x
f x
x
y-int: 3 x-int(s): 4,3 ;
Vertical: 2 2 , x ; Horizontal: 1 y
2
2
2 3
( )
3 10 8
x x
f x
x x
y-int:
3
8
x-int(s):
3
,1
2
;
Vertical:
2
4
3
, x ; Horizontal:
2
3
y
2
2
3
( )
2 10
x
f x
x x
y-int:
3
10
x-int(s): 3, 3 ; Vertical:
5
2
2
, x ;
Horizontal:
1
2
y
Extra Problems-Answers
Answers
Page 356
Page 357
Page 358
Page 359
Page 360
7.4 Ellipses 419
7.4 Ellipses
In the definition of a circle, Definition 7.1, we fixed a point called the center and considered all
of the points which were a fixed distance r from that one point. For our next conic section, the
ellipse, we fix two distinct points and a distance d to use in our definition.
Definition 7.4. Given two distinct points F
1
and F
2
in the plane and a fixed distance d, an
ellipse is the set of all points (x, y) in the plane such that the sum of the distance from F
1
to
(x, y) and the distance from F
2
to (x, y) is d. The points F
1
and F
2
are called the foci
a
of the
ellipse.
a
the plural of ‘focus’
(x, y)
d
1
d
2
F
1
F
2
d
1
+ d
2
= d for all (x, y) on the ellipse
We may imagine taking a length of string and anchoring it to two points on a piece of paper. The
curve traced out by taking a pencil and moving it so the string is always taut is an ellipse.
The center of the ellipse is the midpoint of the line segment connecting the two foci. The major
axis of the ellipse is the line segment connecting two opposite ends of the ellipse which also contains
the center and foci. The minor axis of the ellipse is the line segment connecting two opposite
ends of the ellipse which contains the center but is perpendicular to the major axis. The vertices
of an ellipse are the points of the ellipse which lie on the major axis. Notice that the center is also
the midpoint of the major axis, hence it is the midpoint of the vertices. In pictures we have,
Page 361
Taken from Stitz and Zeager
420 Hooked on Conics
F
1
F
2
V
2
V
1
C
Major Axis
M
i
n
o
r
A
x
i
s
An ellipse with center C; foci F
1
, F
2
; and vertices V
1
, V
2
Note that the major axis is the longer of the two axes through the center, and likewise, the minor
axis is the shorter of the two. In order to derive the standard equation of an ellipse, we assume that
the ellipse has its center at (0, 0), its major axis along the x-axis, and has foci (c, 0) and (−c, 0)
and vertices (−a, 0) and (a, 0). We will label the y-intercepts of the ellipse as (0, b) and (0, −b) (We
assume a, b, and c are all positive numbers.) Schematically,
(−c, 0) (c, 0) (−a, 0) (a, 0)
(0, b)
(0, −b)
(x, y)
x
y
Note that since (a, 0) is on the ellipse, it must satisfy the conditions of Definition 7.4. That is, the
distance from (−c, 0) to (a, 0) plus the distance from (c, 0) to (a, 0) must equal the fixed distance
d. Since all of these points lie on the x-axis, we get
distance from (−c, 0) to (a, 0) + distance from (c, 0) to (a, 0) = d
(a + c) + (a −c) = d
2a = d
Page 362
Taken from Stitz and Zeager
7.4 Ellipses 421
In other words, the fixed distance d mentioned in the definition of the ellipse is none other than
the length of the major axis. We now use that fact (0, b) is on the ellipse, along with the fact that
d = 2a to get
distance from (−c, 0) to (0, b) + distance from (c, 0) to (0, b) = 2a
(0 −(−c))
2
+ (b −0)
2
+
(0 −c)
2
+ (b −0)
2
= 2a
√
b
2
+ c
2
+
√
b
2
+ c
2
= 2a
2
√
b
2
+ c
2
= 2a
√
b
2
+ c
2
= a
From this, we get a
2
= b
2
+c
2
, or b
2
= a
2
−c
2
, which will prove useful later. Now consider a point
(x, y) on the ellipse. Applying Definition 7.4, we get
distance from (−c, 0) to (x, y) + distance from (c, 0) to (x, y) = 2a
(x −(−c))
2
+ (y −0)
2
+
(x −c)
2
+ (y −0)
2
= 2a
(x + c)
2
+ y
2
+
(x −c)
2
+ y
2
= 2a
In order to make sense of this situation, we need to do some rearranging, squaring, and more
rearranging.
1
(x + c)
2
+ y
2
+
(x −c)
2
+ y
2
= 2a
(x + c)
2
+ y
2
= 2a −
(x −c)
2
+ y
2
(x + c)
2
+ y
2
2
=
2a −
(x −c)
2
+ y
2
2
(x + c)
2
+ y
2
= 4a
2
−4a
(x −c)
2
+ y
2
+ (x −c)
2
+ y
2
4a
(x −c)
2
+ y
2
= 4a
2
+ (x −c)
2
−(x + c)
2
4a
(x −c)
2
+ y
2
= 4a
2
−4cx
a
(x −c)
2
+ y
2
= a
2
−cx
a
(x −c)
2
+ y
2
2
=
a
2
−cx
2
a
2
(x −c)
2
+ y
2
= a
4
−2a
2
cx + c
2
x
2
a
2
x
2
−2a
2
cx + a
2
c
2
+ a
2
y
2
= a
4
−2a
2
cx + c
2
x
2
a
2
x
2
−c
2
x
2
+ a
2
y
2
= a
4
−a
2
c
2
a
2
−c
2
x
2
+ a
2
y
2
= a
2
a
2
−c
2
We are nearly finished. Recall that b
2
= a
2
−c
2
so that
a
2
−c
2
x
2
+ a
2
y
2
= a
2
a
2
−c
2
b
2
x
2
+ a
2
y
2
= a
2
b
2
x
2
a
2
+
y
2
b
2
= 1
1
In other words, tons and tons of Intermediate Algebra. Stay sharp, this is not for the faint of heart.
Page 363
Taken from Stitz and Zeager
7.5 Hyperbolas 433
7.5 Hyperbolas
In the definition of an ellipse, Definition 7.4, we fixed two points called foci and looked at points
whose distances to the foci always added to a constant distance d. Those prone to syntactical
tinkering may wonder what, if any, curve we’d generate if we replaced added with subtracted.
The answer is a hyperbola.
Definition 7.6. Given two distinct points F
1
and F
2
in the plane and a fixed distance d, a
hyperbola is the set of all points (x, y) in the plane such that the absolute value of the difference
of the distances between the foci and (x, y) is d. The points F
1
and F
2
are called the foci of the
hyperbola.
(x
1
, y
1
)
(x
2
, y
2
)
F
1
F
2
In the figure above:
the distance from F
1
to (x
1
, y
1
) −the distance from F
2
to (x
1
, y
1
) = d
and
the distance from F
2
to (x
2
, y
2
) −the distance from F
1
to (x
2
, y
2
) = d
Note that the hyperbola has two parts, called branches. The center of the hyperbola is the
midpoint of the line segment connecting the two foci. The transverse axis of the hyperbola is
the line segment connecting two opposite ends of the hyperbola which also contains the center and
foci. The vertices of a hyperbola are the points of the hyperbola which lie on the transverse axis.
In addition, we will show momentarily that there are lines called asymptotes which the branches
of the hyperbola approach for large x and y values. They serve as guides to the graph. In pictures,
Page 364
Taken from Stitz and Zeager
434 Hooked on Conics
V
2
V
1
F
1
F
2
Transverse Axis
C
A hyperbola with center C; foci F
1
, F
2
; and vertices V
1
, V
2
and asymptotes (dashed)
Before we derive the standard equation of the hyperbola, we need to discuss one further parameter,
the conjugate axis of the hyperbola. The conjugate axis of a hyperbola is the line segment
through the center which is perpendicular to the transverse axis and has the same length as the
line segment through a vertex which connects the asymptotes. In pictures we have
V
2
V
1 C
C
o
n
j
u
g
a
t
e
A
x
i
s
Note that in the diagram, we can construct a rectangle using line segments with lengths equal to
the lengths of the transverse and conjugate axes whose center is the center of the hyperbola and
whose diagonals are contained in the asymptotes. This guide rectangle, which is very similar to
the one we created in the Section 7.4 to help us graph ellipses, will aid us in graphing hyperbolas
when the time comes.
Suppose we wish to derive the equation of a hyperbola. For simplicity, we shall assume that the
center is (0, 0), the vertices are (a, 0) and (−a, 0) and the foci are (c, 0) and (−c, 0). We label the
Page 365
Taken from Stitz and Zeager
7.5 Hyperbolas 435
endpoints of the conjugate axis (0, b) and (0, −b). (Although b does not enter into our derivation,
we will have to justify this choice as you shall see later.) As before, we assume a, b, and c are all
positive numbers. Schematically we have
x
y
(a, 0) (−a, 0)
(0, b)
(0, −b)
(−c, 0) (c, 0)
(x, y)
Since (a, 0) is on the hyperbola, it must satisfy the conditions of Definition 7.6. That is, the distance
from (−c, 0) to (a, 0) minus the distance from (c, 0) to (a, 0) must equal the fixed distance d. Since
all these points lie on the x-axis, we get
distance from (−c, 0) to (a, 0) −distance from (c, 0) to (a, 0) = d
(a + c) −(c −a) = d
2a = d
In other words, the fixed distance d from the definition of the hyperbola is actually the length of
the transverse axis! (Where have we seen that type of coincidence before?) Now consider a point
(x, y) on the hyperbola. Applying Definition 7.6, we get
distance from (−c, 0) to (x, y) −distance from (c, 0) to (x, y) = 2a
(x −(−c))
2
+ (y −0)
2
−
(x −c)
2
+ (y −0)
2
= 2a
(x + c)
2
+ y
2
−
(x −c)
2
+ y
2
= 2a
Using the same arsenal of Intermediate Algebra weaponry we used in deriving the standard formula
of an ellipse, Equation 7.4, we arrive at the following.
1
1
It is a good exercise to actually work this out.
Page 366
Taken from Stitz and Zeager
436 Hooked on Conics
a
2
−c
2
x
2
+ a
2
y
2
= a
2
a
2
−c
2
What remains is to determine the relationship between a, b and c. To that end, we note that since
a and c are both positive numbers with a < c, we get a
2
< c
2
so that a
2
−c
2
is a negative number.
Hence, c
2
− a
2
is a positive number. For reasons which will become clear soon, we re-write the
equation by solving for y
2
/x
2
to get
a
2
−c
2
x
2
+ a
2
y
2
= a
2
a
2
−c
2
−
c
2
−a
2
x
2
+ a
2
y
2
= −a
2
c
2
−a
2
a
2
y
2
=
c
2
−a
2
x
2
−a
2
c
2
−a
2
y
2
x
2
=
c
2
−a
2
a
2
−
c
2
−a
2
x
2
As x and y attain very large values, the quantity
c
2
−a
2
x
2
→0 so that
y
2
x
2
→
c
2
−a
2
a
2
. By setting
b
2
= c
2
−a
2
we get
y
2
x
2
→
b
2
a
2
. This shows that y →±
b
a
x as |x| grows large. Thus y = ±
b
a
x are the
asymptotes to the graph as predicted and our choice of labels for the endpoints of the conjugate
axis is justified. In our equation of the hyperbola we can substitute a
2
−c
2
= −b
2
which yields
a
2
−c
2
x
2
+ a
2
y
2
= a
2
a
2
−c
2
−b
2
x
2
+ a
2
y
2
= −a
2
b
2
x
2
a
2
−
y
2
b
2
= 1
The equation above is for a hyperbola whose center is the origin and which opens to the left and
right. If the hyperbola were centered at a point (h, k), we would get the following.
Equation 7.6. The Standard Equation of a Horizontal
a
Hyperbola For positive numbers
a and b, the equation of a horizontal hyperbola with center (h, k) is
(x −h)
2
a
2
−
(y −k)
2
b
2
= 1
a
That is, a hyperbola whose branches open to the left and right
If the roles of x and y were interchanged, then the hyperbola’s branches would open upwards and
downwards and we would get a ‘vertical’ hyperbola.
Equation 7.7. The Standard Equation of a Vertical Hyperbola For positive numbers a
and b, the equation of a vertical hyperbola with center (h, k) is:
(y −k)
2
b
2
−
(x −h)
2
a
2
= 1
The values of a and b determine how far in the x and y directions, respectively, one counts from the
center to determine the rectangle through which the asymptotes pass. In both cases, the distance
Page 367
Taken from Stitz and Zeager
7.5 Hyperbolas 437
from the center to the foci, c, as seen in the derivation, can be found by the formula c =
√
a
2
+ b
2
.
Lastly, note that we can quickly distinguish the equation of a hyperbola from that of a circle or
ellipse because the hyperbola formula involves a difference of squares where the circle and ellipse
formulas both involve the sum of squares.
Example 7.5.1. Graph the equation
(x −2)
2
4
−
y
2
25
= 1. Find the center, the lines which contain
the transverse and conjugate axes, the vertices, the foci and the equations of the asymptotes.
Solution. We first see that this equation is given to us in the standard form of Equation 7.6.
Here x−h is x−2 so h = 2, and y −k is y so k = 0. Hence, our hyperbola is centered at (2, 0). We
see that a
2
= 4 so a = 2, and b
2
= 25 so b = 5. This means we move 2 units to the left and right
of the center and 5 units up and down from the center to arrive at points on the guide rectangle.
The asymptotes pass through the center of the hyperbola as well as the corners of the rectangle.
This yields the following set up.
x
y
−2 −1 1 2 3 4 5 6
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
7
Since the y
2
term is being subtracted from the x
2
term, we know that the branches of the hyperbola
open to the left and right. This means that the transverse axis lies along the x-axis. Hence, the
conjugate axis lies along the vertical line x = 2. Since the vertices of the hyperbola are where the
hyperbola intersects the transverse axis, we get that the vertices are 2 units to the left and right of
(2, 0) at (0, 0) and (4, 0). To find the foci, we need c =
√
a
2
+ b
2
=
√
4 + 25 =
√
29. Since the foci
lie on the transverse axis, we move
√
29 units to the left and right of (2, 0) to arrive at (2 −
√
29, 0)
(approximately (−3.39, 0)) and (2 +
√
29, 0) (approximately (7.39, 0)). To determine the equations
of the asymptotes, recall that the asymptotes go through the center of the hyperbola, (2, 0), as well
as the corners of guide rectangle, so they have slopes of ±
b
a
= ±
5
2
. Using the point-slope equation
of a line, Equation 2.2, yields
Page 368
Taken from Stitz and Zeager
438 Hooked on Conics
y = ±
5
2
(x −2) + 0,
so we get y =
5
2
x −5 and y = −
5
2
x + 5. Putting it all together, we get
x
y
−3 −2 −1 1 2 3 4 5 6 7
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
7
Example 7.5.2. Find the equation of the hyperbola with asymptotes y = ±2x and vertices (±5, 0).
Solution. Plotting the data given to us, we have
x
y
−5 5
−5
5
This graph not only tells us that the branches of the hyperbola open to the left and to the right,
it also tells us that the center is (0, 0). Hence, our standard form is
x
2
a
2
−
y
2
b
2
= 1.
Page 369
Taken from Stitz and Zeager
Exercises 4.9—Ellipse and Hyperbola
In exercises 1-16, find the vertices and foci for each ellipse. Graph each
ellipse.
1.
2 2
1
16 4
x y
2.
2 2
1
25 16
x y
3.
2 2
1
9 36
x y
4.
2 2
1
16 49
x y
5.
2 2
1
25 16
x y
6.
2 2
1
49 36
x y
7.
2 2
1
49 81
x y
8.
2 2
1
64 100
x y
9.
2 2
1
81 25
4 16
x y
10.
2 2
1
9 25
4 4
x y
11.
2 2
25 4 100 x y 12.
2 2
9 4 36 x y
13.
2 2
4 49 196 x y 14.
2 2
4 25 100 x y
15.
2 2
2 4 x y 16.
2 2
8 9 72 x y
In exercises 17-20, find the standard form of the equation of each ellipse
and give the location of its foci.
17. 18.
19. 20.
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
Page 370
Ellipses
In exercises 21-32, find the standard form of the equation of each ellipse
satisfying the given information.
21. Ver t i ces:
30 , ; mi nor axi s of l engt h 2.
22. Ver t i ces:
0 5 , ; mi nor axi s of l engt h 4.
23. Ver t i ces:
0 4 , ; and passi ng t hr ough t he poi nt 13 ( , ) .
24. Ver t i ces:
30 , ; and passi ng t hr ough t he poi nt 21 ( , ) .
25. Maj or axi s i s hor i zont al wi t h l engt h 8; l engt h of mi nor axi s =4;
cent er i s 0 0 ( , )
26. Maj or axi s i s hor i zont al wi t h l engt h 12; l engt h of mi nor axi s =6;
cent er i s 0 0 ( , )
27. Ver t i ces:
80 , ; Foci :
50 ,
28. Ver t i ces:
60 , ; Foci :
20 ,
29. Ver t i ces:
0 7 , ; Foci :
0 4 ,
30. Ver t i ces:
0 4 , ; Foci :
0 3 ,
31. Foci :
20 , ; y- i nt er cept s: 3 and 3
32. Foci :
0 2 , ; x- i nt er cept s: 2 and 2
In exercises 33-46, find the vertices, asymptotes, and foci for each
hyperbola. Graph each hyperbola.
33.
2 2
1
9 25
x y
34.
2 2
1
16 25
x y
35.
2 2
1
100 64
x y
36.
2 2
1
144 81
x y
37.
2 2
1
25 64
y x
38.
2 2
1
9 64
y x
39.
2 2
1
49 16
y x
40.
2 2
1
100 5
y x
41.
2 2
4 36 y x 42.
2 2
16 64 y x
43.
2 2
9 4 36 x y 44.
2 2
4 25 100 x y
45.
2 2
9 25 225 y x 46.
2 2
16 9 144 x y
Page 371
Hyperbolas
In exercises 47-50, find the standard form of the equation of each hyperbola
and give the location of its foci.
47. 48.
49. 50.
In exercises 51-58, find the standard form of the equation of each hyperbola
satisfying the given information.
51. Ver t i ces:
30 , ; Foci :
40 ,
52. Ver t i ces:
50 , ; Foci :
7 0 ,
53. Ver t i ces:
0 1 , ; Foci :
0 3 ,
54. Ver t i ces:
0 2 , ; Foci :
0 6 ,
55. Foci :
60 , ; Endpoi nt s of conj ugat e axi s:
0 3 ,
56. Foci :
0 4 , ; Endpoi nt s of conj ugat e axi s:
20 ,
57. Tr ansver se axi s i s hor i zont al wi t h l engt h 8;
l engt h of conj ugat e axi s =4; cent er i s 00 ( , )
58. Tr ansver se axi s i s ver t i cal wi t h l engt h 6;
l engt h of conj ugat e axi s =14; cent er i s 00 ( , )
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-4
-3
-2
-1
1
2
3
4
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
Page 372
4.9—Ellipse and Hyperbola
1. Ver t i ces:
( )
4 0 ± , ; 3. Ver t i ces:
( )
0 6 ± , ; 5. Ver t i ces:
( )
50 ± , ;
Foci :
( )
2 3 0 ± , Foci :
( )
0 3 3 ± , Foci :
( )
30 ± ,
7. Ver t i ces: ( )
0 9 ± , ; 9. Ver t i ces:
9
0
2
| |
±
|
\ .
, ; 11. Ver t i ces:
( )
0 5 ± , ;
Foci :
( )
0 4 2 ± , Foci :
299
0
4
| |
±
|
|
\ .
, Foci :
( )
0 21 ± ,
13. Ver t i ces:
( )
7 0 ± , ; 15. Ver t i ces:
( )
0 2 ± , ; 17.
2
2
1
4
+ =
x
y ; Foci :
( )
3 0 ± ,
Foci :
( )
3 5 0 ± , Foci :
( )
0 2 ± , 19.
2 2
1
9 25
+ =
x y
; Foci :
( )
0 4 ± ,
21.
2 2
1
9 1
+ =
x y
23.
2 2
1
16
16
7
+ =
x y
25.
2 2
1
16 4
+ =
x y
-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-7-6-5-4-3-2-1 1 2 3 4 5 6 7
-8
-6
-4
-2
2
4
6
8
-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-5 -4 -3 -2 -1 1 2 3 4 5
-2
-1
1
2
-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8
-10
-8
-6
-4
-2
2
4
6
8
10
-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Page 373
Answers
27.
2 2
1
64 39
+ =
x y
29.
2 2
1
33 49
+ =
x y
31.
2 2
1
13 9
+ =
x y
33. Ver t i ces:
( )
30 ± , ; 35. Ver t i ces:
( )
10 0 ± , ; 37. Ver t i ces:
( )
0 5 ± , ;
Foci :
( )
34 0 ± , Foci :
( )
2 41 0 ± , Foci :
( )
0 89 ± ,
Asympt ot es:
5
3
= ± y x Asympt ot es:
4
5
= ± y x Asympt ot es:
5
8
= ± y x
39. Ver t i ces:
( )
0 7 ± , ; 41. Ver t i ces:
( )
0 3 ± , ; 43. Ver t i ces:
( )
20 ± , ;
Foci :
( )
0 65 ± , Foci :
( )
0 3 5 ± , Foci :
( )
13 0 ± ,
Asympt ot es:
7
4
= ± y x Asympt ot es:
1
2
= ± y x Asympt ot es:
3
2
= ± y x
45. Ver t i ces: ( )
0 5 ± , ; 47.
2 2
1
9 4
÷ =
x y
; Foci :
( )
13 0 ± ,
Foci :
( )
0 34 ± , 49.
2 2
1
16 9
÷ =
y x
; Foci :
( )
0 5 ± ,
Asympt ot es:
5
3
= ± y x 51.
2 2
1
9 7
÷ =
x y
55.
2 2
1
27 9
÷ =
x y
53.
2 2
1
1 8
÷ =
y x
57.
2 2
1
16 4
÷ =
x y
-5 -4 -3 -2 -1 1 2 3 4 5
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-12-10-8 -6 -4 -2 2 4 6 8 1012
-10
-8
-6
-4
-2
2
4
6
8
10
-10 -8 -6 -4 -2 2 4 6 8 10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
-10 -8 -6 -4 -2 2 4 6 8 10
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
45.
Page 374
Page 375
Page 376
Page 377
Page 378
Page 379
Page 380
Page 381
Page 382
Page 383
Page 384
Page 385
Page 386
Page 387
Page 388
Page 389
Page 390
Page 391
Exercises 5.1 and 5.2—Exponential Functions
I n exer ci ses 1- 8, gr aph each f unct i on by maki ng a t abl e of coor di nat es.
1. 5 = ( )
x
f x 2. 4 = ( )
x
f x 3.
1
3
| |
=
|
\ .
( )
x
f x
4.
1
2
| |
=
|
\ .
( )
x
f x 5.
3
2
| |
=
|
\ .
( )
x
f x 6.
4
3
| |
=
|
\ .
( )
x
f x
7.
( )
06 = ( ) .
x
f x 8.
( )
09 = ( ) .
x
f x
By t r ansl at i ng, r ef l ect i ng, and st r et chi ng t he gr aph of 2 = ( )
x
f x , obt ai n t he
gr aphs of t he f ol l owi ng f unct i ons. Gi ve t he domai n, r ange, and equat i on of
any asympt ot es of t he f unct i on.
9.
1
2
+
= ( )
x
f x 10.
2
2
÷
= ( )
x
f x 11. 2 2 = + ( )
x
f x
12. 2 1 = ÷ ( )
x
f x 13.
2
2 3
÷
= ÷ ( )
x
f x 14.
1
2 3
+
= + ( )
x
f x
15. 2
÷
= ( )
x
f x 16. 2 1
÷
= + ( )
x
f x 17. 2 = ÷ ( )
x
f x
18.
1
2
÷
= ÷ ( )
x
f x 19.
1
2
+
= ÷ ( )
x
f x 20.
1
2 3
+
= ÷ + ( )
x
f x
21.
1
2 3
2
= ÷ ( )
x
f x 22.
1
1
2
2
+
= ( )
x
f x 23.
1
2 2 1
÷
= + ( )
x
f x
By t r ansl at i ng, r ef l ect i ng, and st r et chi ng t he gr aph of 3 = ( )
x
f x , obt ai n t he
gr aphs of t he f ol l owi ng f unct i ons. Gi ve t he domai n, r ange, and equat i on of
any asympt ot es of t he f unct i on.
24.
2
3
÷
= ( )
x
f x 25.
1
3
+
= ( )
x
f x 26. 3 1 = ÷ ( )
x
f x
27. 3 2 = + ( )
x
f x 28.
1
3 3
+
= + ( )
x
f x 29.
2
3 3
÷
= ÷ ( )
x
f x
30. 3
÷
= ( )
x
f x 31. 3 1
÷
= + ( )
x
f x 32. 3 = ÷ ( )
x
f x
33.
1
3
+
= ÷ ( )
x
f x 34.
1
3
÷
= ÷ ( )
x
f x 35.
1
3 2
+
= ÷ + ( )
x
f x
36.
1
3 3
2
= ÷ ( )
x
f x 37.
1
23
+
= ( )
x
f x 38.
1
23 1
÷
= + ( )
x
f x
By t r ansl at i ng, r ef l ect i ng, and st r et chi ng t he gr aph of = ( )
x
f x e , obt ai n t he
gr aphs of t he f ol l owi ng f unct i ons. Gi ve t he domai n, r ange, and equat i on of
any asympt ot es of t he f unct i on.
39.
1 ÷
= ( )
x
f x e 40.
2 +
= ( )
x
f x e 41. 2 = ÷ ( )
x
f x e
42. 1 = + ( )
x
f x e 43.
1
2
+
= + ( )
x
f x e 44.
2
1
÷
= ÷ ( )
x
f x e
45.
÷
= ( )
x
f x e 46. 2
÷
= + ( )
x
f x e 47. = ÷ ( )
x
f x e
48.
1 ÷
= ÷ ( )
x
f x e 49.
2 +
= ÷ ( )
x
f x e 50.
2
1
+
= ÷ ÷ ( )
x
f x e
51. 2 3 = ÷ ( )
x
f x e 52.
1
1
2
+
= ( )
x
f x e 53.
1
1
1
2
÷
= + ( )
x
f x e
Page 392
I n exer ci ses 54- 77, sol ve each exponent i al equat i on i n by expr essi ng each
si de as a power of t he same base and t hen equat i ng t he exponent s.
54. 4 8
x
= 55.
1
3 81
x+
=
56.
1
2
8
x
=
57.
3 1
1
10
100
x+
=
58.
2
5 125
x
= 59. =
2
3 81
x
60.
3 5
4 8
x x +
= 61.
4 1
16 4
x÷
= 62.
1
36
6
x
| |
=
|
\ .
63.
1
1
25
5
x ÷
= 64.
2 1
2
1
x
e
e
÷
=
65.
2 x
e e
÷
=
66.
1
4
9 27
x
= 67.
÷
=
1
2
8 4
x
x
68.
2 8 3 1
2 8
x x + ÷
=
69.
2
2
3 81
x
= 70.
6 4
125 0.2
x+
=
71. = 9 3
x x
72.
+
=
1
9 27
x
73.
( )
÷
=
3
4 6 6
x
74.
( )
÷
=
2
6 7 7
x
75. =
1
4
2
x
76. =
3
1
9
3
x
77.
+
=
4
2
1
x
x
e
e
I n exer ci ses 78- 86, sol ve t he f ol l owi ng by usi ng t he appr opr i at e f or mul as.
0
1
| |
= +
|
\ .
( )
nt
r
A t A
n
or
0
= ( )
rt
A t A e
78. I f Manhat t an I sl and had been pur chased f r omNat i ve Amer i cans i n 1626 f or
$24, and i f t hat $24 was i mmedi at el y deposi t ed i n an account t hat pai d
i nt er est at t he r at e of 5%compounded year l y, what woul d be t he val ue of
t he i nvest ment i n 1998? What woul d be t he val ue of t he i nvest ment i f
t he i nt er est r at e wer e 6%?
79. Fi nd t he val ue of a $5000 i nvest ment f or 10 year s at an annual i nt er est
r at e of 6%i f i nt er est i s compounded:
a) annual l y
b) mont hl y
c) dai l y
d) each mi nut e
e) cont i nuousl y
80. Fi nd t he val ue on an i nvest ment af t er 8 year s i f $3500 i s i nvest ed at an
annual i nt er est r at e of 4. 5%compounded quar t er l y.
Page 393
81. Fi nd t he val ue on an i nvest ment af t er 30 mont hs i f $1200 i s i nvest ed at
an annual i nt er est r at e of 5. 25%compounded dai l y.
82. An i nvest or wi t h $2000 t o i nvest f or 3 year s may i nvest t hi s money at an
annual r at e of 8%compounded mont hl y or at an annual r at e of 6%
compounded cont i nuousl y. Whi ch st r at egy i s bet t er ? How much mor e money
i s made by f ol l owi ng t he bet t er st r at egy?
83. Fi nd t he val ue on an i nvest ment af t er 8 year s i f $1500 i s i nvest ed at an
annual i nt er est r at e of 4%compounded cont i nuousl y.
84. Fi nd t he val ue on an i nvest ment af t er 5 year s i f $10, 000 i s i nvest ed at
an annual i nt er est r at e of 8%compounded cont i nuousl y.
85. How much must be i nvest ed i ni t i al l y at 5%compounded cont i nuousl y i f t he
val ue of t he i nvest ment i s t o be $4, 200 af t er 7 year s?
86. Det er mi ne t he annual i nt er est r at e r i f an i ni t i al i nvest ment of $1000
i s t o gr ow t o $1, 250 i n 3 year s i f i nt er est i s compounded mont hl y.
Page 394
Section 5.1 and 5.2—Exponential Functions
1. 3. 5.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
7. 9. D:( , ) , R: (0, ) 11. D:( , ) , R: (2, )
Asymptote: 0 y Asymptote: 2 y
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
13. D:( , ) , R: ( 3, ) 15. D:( , ) , R: (0, ) 17. D:( , ) , R: ( ,0)
Asymptote: 3 y Asymptote: 0 y Asymptote: 0 y
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Page 395
19. D:( , ) ÷· · , R: ( ,0) ÷· 21. D:( , ) ÷· · , R: ( 3, ) ÷ · 23. D:( , ) ÷· · , R: (1, ) ·
Asymptote: 0 y = Asymptote: 3 y = ÷ Asymptote: 1 y =
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-6
-5
-4
-3
-2
-1
1
2
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
25. D:( , ) ÷· · , R: (0, ) · 27. D:( , ) ÷· · , R: (2, ) · 29. D:( , ) ÷· · , R: ( 3, ) ÷ ·
Asymptote: 0 y = Asymptote: 2 y = Asymptote: 3 y = ÷
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
31. D:( , ) ÷· · , R: (1, ) · 33. D:( , ) ÷· · , R: (0, ) · 35. D:( , ) ÷· · , R: ( ,2) ÷·
Asymptote: 1 y = Asymptote: 0 y = Asymptote: 2 y =
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Page 396
37. D:( , ) ÷· · , R: (0, ) · 39. D:( , ) ÷· · , R: (0, ) · 41. D:( , ) ÷· · , R: ( 2, ) ÷ ·
Asymptote: 0 y = Asymptote: 0 y = Asymptote: 2 y = ÷
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
43. D:( , ) ÷· · , R: (2, ) · 45. D:( , ) ÷· · , R: (0, ) · 47. D:( , ) ÷· · , R: ( ,0) ÷·
Asymptote: 2 y = Asymptote: 0 y = Asymptote: 0 y =
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
49. D:( , ) ÷· · , R: ( ,0) ÷· 51. D:( , ) ÷· · , R: ( 3, ) ÷ · 53. D:( , ) ÷· · , R: (1, ) ·
Asymptote: 0 y = Asymptote: 3 y = ÷ Asymptote: 1 y =
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-6
-5
-4
-3
-2
-1
1
2
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
Page 397
55. { } 3
57. { } 1 ÷
59. { } 2
61.
3
8
¦ ¹
´ `
¹ )
63.
3
2
¦ ¹
´ `
¹ )
65.
5
2
¦ ¹
´ `
¹ )
67. { } 4
69.
{ }
2 ±
71. { } 0,4
73. { } 5
75.
1
4
÷ ¦ ¹
´ `
¹ )
77.
4
3
÷ ¦ ¹
´ `
¹ )
79. a) 895424 $ . b) 909698 $ . c) 911014 $ . d) 911058 $ . e) 911060 $ . f) 911060 $ .
80. 500658 $ .
82. Compounded monthly, 14604 $ . more
84. 1491825 $ .
86. 75 . %
Page 398
Page 399
Page 400
Page 401
Page 402
Page 403
Page 404
Page 405
Page 406
Page 407
Page 408
Page 409
Section 5.3—Logarithms
1. 100000 5 log( , ) =
3.
3
1
2
9
log
⎛ ⎞
= −
⎜ ⎟
⎝ ⎠
5.
( )
36
1
6
2
log =
7.
( )
log
e
P t = or
( )
ln P t =
9.
2
1 1
2 4
=
11.
1
1
10
10
−
=
13.
0
1 e =
15. 3
N
t =
17. 0
19. 3 −
21. 2 −
23. 12
25.
1
2
27.
1
9
27. 1857 .
29. 223 . −
31. 0861 .
33. 1365 .
35. a)
3
27 3 log = and
4
16 2 log =
b)
6
40 20588 log . = and
4
16 2 log =
37. a) 243 x = b) 2 x = c) 21 , x = −
39. a) 08 0969 log . . = − ; 80 19031 log . = ;
b) 10 times
41. a) 51293 . b) 10995 . c) 29887 .
d) 79367 . − e) 44453 . −
45.
1
3
2
(log log ) log
a a a
x y z + −
47. 2 4 3 6 ( log log ) (log log )
a a a a
w z x + − +
49.
1
1
3
3
4
(log log ) ( log log )
a a a a
x y z w + − +
51.
2
1
2 4 1
2
(log ( )) log ( )
a a
x x + − +
53.
2
1 log ( )
a
x x +
55.
4 3
log
a
y x
z
57.
1 1
log
( )( )
a
x
x x + −
Page 410
Answers
342 Exponential and Logarithmic Functions
6.1.1 Exercises
1. Evaluate the expression.
(a) log
3
(27)
(b) log
6
(216)
(c) log
2
(32)
(d) log
6
1
36
(e) log
8
(4)
(f) log
36
(216)
(g) log1
5
(625)
(h) log1
6
(216)
(i) log
36
(36)
(j) log
1
1000000
(k) log(0.01)
(l) ln
e
3
(m) log
4
(8)
(n) log
6
(1)
(o) log
13
√
13
(p) log
36
4
√
36
(q) 7
log
7
(3)
(r) 36
log
36
(216)
(s) log
36
36
216
(t) ln
e
5
(u) log
9
√
10
11
(v) log
3
√
10
5
(w) ln
1
√
e
(x) log
5
3
log
3
(5)
(y) log
e
ln(100)
2. Find the domain of the function.
(a) f(x) = ln(x
2
+ 1)
(b) f(x) = log
7
(4x + 8)
(c) f(x) = ln(4x −20)
(d) f(x) = log
x
2
+ 9x + 18
(e) f(x) = log
x + 2
x
2
−1
(f) f(x) = log
x
2
+ 9x + 18
4x −20
(g) f(x) = ln(7 −x) + ln(x −4)
(h) f(x) = ln(4x −20) + ln
x
2
+ 9x + 18
(i) f(x) = log
x
2
+ x + 1
(j) f(x) =
4
log
4
(x)
(k) f(x) = log
9
(|x + 3| −4)
(l) f(x) = ln(
√
x −4 −3)
(m) f(x) =
1
3 −log
5
(x)
(n) f(x) =
√
−1 −x
log1
2
(x)
(o) f(x) = ln(−2x
3
−x
2
+ 13x −6)
3. For each function given below, find its inverse from the ‘procedural perspective’ discussed in
Example 6.1.5 and graph the function and its inverse on the same set of axes.
(a) f(x) = 3
x+2
−4
(b) f(x) = log
4
(x −1)
(c) f(x) = −2
−x
+ 1
(d) f(x) = 5 log(x) −2
4. Show that log
b
1 = 0 and log
b
b = 1 for every b > 0, b = 1.
5. (Crazy bonus question) Without using your calculator, determine which is larger: e
π
or π
e
.
6. (The Logarithmic Scales) There are three widely used measurement scales which involve
common logarithms: the Richter scale, the decibel scale and the pH scale. The computations
involved in all three scales are nearly identical so pay close attention to the subtle differences.
Page 411
Taken from Stitz and Zeager
6.1 Introduction to Exponential and Logarithmic Functions 345
6.1.2 Answers
1. (a) log
3
(27) = 3
(b) log
6
(216) = 3
(c) log
2
(32) = 5
(d) log
6
1
36
= −2
(e) log
8
(4) =
2
3
(f) log
36
(216) =
3
2
(g) log1
5
(625) = −4
(h) log1
6
(216) = −3
(i) log
36
(36) = 1
(j) log
1
1000000
= −6
(k) log(0.01) = −2
(l) ln
e
3
= 3
(m) log
4
(8) =
3
2
(n) log
6
(1) = 0
(o) log
13
√
13
=
1
2
(p) log
36
4
√
36
=
1
4
(q) 7
log
7
(3)
= 3
(r) 36
log
36
(216)
= 216
(s) log
36
36
216
= 216
(t) ln(e
5
) = 5
(u) log
9
√
10
11
=
11
9
(v) log
3
√
10
5
=
5
3
(w) ln
1
√
e
= −
1
2
(x) log
5
3
log
3
5
= 1
(y) log
e
ln(100)
= 2
2. (a) (−∞, ∞)
(b) (−2, ∞)
(c) (5, ∞)
(d) (−∞, −6) ∪ (−3, ∞)
(e) (−2, −1) ∪ (1, ∞)
(f) (−6, −3) ∪ (5, ∞)
(g) (4, 7)
(h) (5, ∞)
(i) (−∞, ∞)
(j) [1, ∞)
(k) (−∞, −7) ∪ (1, ∞)
(l) (13, ∞)
(m) (0, 125) ∪ (125, ∞)
(n) No domain
(o) (−∞, −3) ∪
1
2
, 2
3. (a) f(x) = 3
x+2
−4
f
−1
(x) = log
3
(x + 4) −2
x
y
y = f(x) = 3
x+2
−4
y = f
−1
(x) = log
3
(x + 4) −2
−4−3−2−1 1 2 3 4 5 6
−4
−3
−2
−1
1
2
3
4
5
6
(b) f(x) = log
4
(x −1)
f
−1
(x) = 4
x
+ 1
x
y
y = f(x) = log
4
(x −1)
y = f
−1
(x) = 4
x
+ 1
−2−1 1 2 3 4 5 6
−2
−1
1
2
3
4
5
6
Page 412
Taken from Stitz and Zeager
6.2 Properties of Logarithms 355
6.2.1 Exercises
1. Expand the following using the properties of logarithms and simplify. Assume when necessary
that all quantities represent positive real numbers.
(a) ln(x
3
y
2
)
(b) log
2
128
x
2
+ 4
(c) log
5
z
25
3
(d) log(1.23 ×10
37
)
(e) ln
√
z
xy
(f) log
5
x
2
−25
(g) log
√
2
4x
3
(h) log1
3
(9x(y
3
−8))
(i) log
1000x
3
y
5
(j) log
3
x
2
81y
4
(k) ln
4
xy
ez
(l) log
6
216
x
3
y
4
(m) ln
3
√
x
10
√
yz
2. Use the properties of logarithms to write the following as a single logarithm.
(a) 4 ln(x) + 2 ln(y)
(b) 3 −log(x)
(c) log
2
(x) + log
2
(y) −log
2
(z)
(d) log
3
(x) −2 log
3
(y)
(e)
1
2
log
3
(x) −2 log
3
(y) −log
3
(z)
(f) 2 ln(x) −3 ln(y) −4 ln(z)
(g) log(x) −
1
3
log(z) +
1
2
log(y)
(h) −
1
3
ln(x) −
1
3
ln(y) +
1
3
ln(z)
(i) log
2
(x) + log1
2
(x −1)
(j) log
2
(x) + log
4
(x −1)
(k) log
5
(x) −3
(l) log
7
(x) + log
7
(x −3) −2
(m) ln(x) +
1
2
3. Use an appropriate change of base formula to convert the following expressions to ones with
the indicated base.
(a) 7
x−1
to base e
(b) log
3
(x + 2) to base 10
(c)
2
3
x
to base e
(d) log(x
2
+ 1) to base e
4. Use the appropriate change of base formula to approximate the following logarithms.
(a) log
3
(12)
(b) log
5
(80)
(c) log
6
(72)
(d) log
4
1
10
(e) log3
5
(1000)
(f) log2
3
(50)
Page 413
Taken from Stitz and Zeager
6.2 Properties of Logarithms 357
6.2.2 Answers
1. (a) 3 ln(x) + 2 ln(y)
(b) 7 −log
2
(x
2
+ 4)
(c) 3 log
5
(z) −6
(d) log(1.23) + 37
(e)
1
2
ln(z) −ln(x) −ln(y)
(f) log
5
(x −5) + log
5
(x + 5)
(g) 3 log
√
2
(x) + 4
(h) −2+log1
3
(x)+log1
3
(y−2)+log1
3
(y
2
+2y+4)
(i) 3 + 3 log(x) + 5 log(y)
(j) 2 log
3
(x) −4 −4 log
3
(y)
(k)
1
4
ln(x) +
1
4
ln(y) −
1
4
−
1
4
ln(z)
(l) 12 −12 log
6
(x) −4 log
6
(y)
(m)
1
3
ln(x) −ln(10) −
1
2
ln(y) −
1
2
ln(z)
2. (a) ln(x
4
y
2
)
(b) log
1000
x
(c) log
2
xy
z
(d) log
3
x
y
2
(e) log
3
√
x
y
2
z
(f) ln
x
2
y
3
z
4
(g) log
x
√
y
3
√
z
(h) ln
3
z
xy
(i) log
2
x
x −1
(j) log
2
x
√
x −1
(k) log
5
x
125
(l) log
7
x(x −3)
49
(m) ln (x
√
e)
3. (a) 7
x−1
= e
(x−1) ln(7)
(b) log
3
(x + 2) =
log(x + 2)
log(3)
(c)
2
3
x
= e
xln(
2
3
)
(d) log(x
2
+ 1) =
ln(x
2
+ 1)
ln(10)
4. (a) log
3
(12) ≈ 2.26186
(b) log
5
(80) ≈ 2.72271
(c) log
6
(72) ≈ 2.38685
(d) log
4
1
10
≈ −1.66096
(e) log3
5
(1000) ≈ −13.52273
(f) log2
3
(50) ≈ −9.64824
Page 414
Taken from Stitz and Zeager
Page 415
Page 416
Page 417
Page 418
Page 419
Page 420
Exercises 5.4—Logarithmic Functions
I n exer ci ses 1- 8, sket ch t he gr aphs of each pai r of f unct i ons on t he same set
of axes. Label al l asympt ot es.
1. 5 = ( )
x
f x and
5
= ( ) log g x x
2. 4 = ( )
x
f x and
4
= ( ) log g x x
3.
1
4
| |
=
|
\ .
( )
x
f x and
1
4
= ( ) log g x x
4.
1
2
| |
=
|
\ .
( )
x
f x and
1
2
= ( ) log g x x
5. = ( )
x
f x e and = ( ) ln g x x
6. 10 = ( )
x
f x and = ( ) log g x x
By t r ansl at i ng, r ef l ect i ng, and st r et chi ng t he gr aph of = ( ) log f x x , obt ai n
t he gr aphs of t he f ol l owi ng f unct i ons. Gi ve t he domai n, r ange, and equat i on
of any asympt ot es of t he f unct i on.
7. 1 = ÷ ( ) log( ) f x x 8. 2 = + ( ) log( ) f x x
9. 1 = + ( ) log f x x 10. 2 = ÷ ( ) log f x x
11. 2 3 = + ÷ ( ) log( ) f x x 12. 1 4 = ÷ + ( ) log( ) f x x
13. = ÷ ( ) log f x x 14. 2 = ( ) log f x x
15. 1 = ÷ ( ) log f x x 16. 2 = + ( ) log f x x
17. 2 1 = ÷ ÷ ( ) log( ) f x x 18. 3 2 = ÷ + ( ) log( ) f x x
By t r ansl at i ng, r ef l ect i ng, and st r et chi ng t he gr aph of = ( ) ln f x x , obt ai n t he
gr aphs of t he f ol l owi ng f unct i ons. Gi ve t he domai n, r ange, and equat i on of
any asympt ot es of t he f unct i on.
19. ( ) ( 1) f x ln x = + 20. ( ) ( 3) f x ln x = ÷
21. 4 f ( x ) ln x = + 22. ( ) 3 f x lnx = ÷
23. ( ) ( 2) 1 f x ln x = ÷ + 24. ( ) ( 2) 4 f x ln x = + ÷
25. = ÷ ( ) ln f x x 26.
1
2
= ( ) log f x x
27. 3 2 = ÷ + ( ) ln( ) f x x 28. 2 1 = ÷ ÷ ( ) ln( ) f x x
Page 421
Section 5.4—Logarithmic Functions
1. 3. 5.
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
7. D:(1, ) , R: ( , ) 9. D: (0, ) , R: ( , ) 11. D:( 2, ) , R: ( , )
Asymptote: 1 x Asymptote: 0 x Asymptote: 2 x
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
13. D: (0, ) , R: ( , ) 15. D: (0, ) , R: ( , ) 17. D:(1, ) , R: ( , )
Asymptote: 0 x Asymptote: 0 x Asymptote: 1 x
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
Page 422
Answers
19. D:( 1, ) , R: ( , ) 21. D:(0, ) , R: ( , ) 23. D:(2, ) , R: ( , )
Asymptote: 1 x Asymptote: 0 x Asymptote: 2 x
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
25. D: (0, ) , R: ( , ) 27. D: ( 2, ) , R: ( , )
Asymptote: 0 x Asymptote: 2 x
-2 -1 1 2 3 4 5 6
-4
-3
-2
-1
1
2
3
4
-2 -1 1 2 3 4 5 6
-2
-1
1
2
3
4
5
6
Page 423
Page 424
Page 425
Page 426
Page 427
Page 428
Page 429
Page 430
Page 431
The second half of this video
is for these exercises.
Page 432
Section 5.5—Exponential and Logarithmic Equations
1. a) 1 x =
b)
3
0
2
, x =
c) 03 , x =
d) 0 x =
3. 18 . x ≈
5. 058 . x ≈ −
7. 120 . x ≈ − or
( )
ln P t =
9. 029 . x ≈
11. 0 and 281 . x x = ≈
13. 018 . x ≈ −
15. 021 . x ≈
17.
1
216
x =
19.
1
10
x =
21. 3 x =
23. 4 x =
25.
9
2
x =
27. 1 x = −
29. 4 x =
31. 5 x = Zero is an extraneous solution
33.
6
5
x =
35.
3
2
x =
37. 1 and 10000 x =
39. 10000 x =
41. 2 x =
Page 433
Answers
Exponential Equations
Solve. Round answers to 3 decimal places.
1. 4 21
x
=
2. 7 35
x
=
3.
5
2 11
x
=
4.
1
7 20
x+
=
5.
1
3 16
x+
=
6.
3 2
5 7
x+
=
7.
2
9 17 6
x÷
÷ =
8.
4 1
11
x
e
+
=
9. 9 107
x
e =
10.
5
3 25
x
e =
11.
2 3
4 120
x
e
÷
=
12.
3 4
1000 3000
x
e
÷
=
13.
4 1
3 1 19
x
e
+
+ =
14.
2
3(2 9 ) 11
x
+ =
15.
3 4
23
x
e
÷
=
16.
1 2 5
3 7
x x + ÷
=
17.
8 2 5 2
2 3
x x + ÷
=
18.
3 1 5 4
5 7
x x + +
=
Page 434
The second half of the video
is for these exercises.
Exponential Equations-Answers
1.
ln21
ln4
x = ~2.196
2.
ln35
ln7
x = ~ 1.827
3.
ln11
5ln2
x = ~ 0.692
4.
ln20
1
ln7
x = ÷ ~ 0.540
5.
ln16
1
ln3
x = ÷ ~1.524
6.
ln7 2ln5
3ln5
x
÷
= 0.264 ~ ÷
7.
ln23
2
ln9
x = + ~3.427
8.
1 ln11
4
x
÷ +
= ~0.349
9.
107
ln
9
x
| |
=
|
\ .
~2.476
10.
1 25
ln
5 3
x
| |
=
|
\ .
~0.424
11.
3 ln30
2
x
+
= ~3.201
12.
4 ln3
3
x
+
= ~1.700
13.
1 ln6
4
x
÷ +
= ~0.198
14.
( )
5
ln
3
2ln9
x = ~0.116
15.
3 ln23
4
x
÷
= ~ ÷ 0.034
16.
ln3 5ln7
ln3 2ln7
x
÷ ÷
=
÷
~3.877
17.
2ln3 2ln2
8ln2 5ln3
x
÷ ÷
=
÷
~ ÷ 68.760
18.
4ln7 ln5
3ln5 5ln7
x
÷
=
÷
~ ÷ 1.260
Page 435
Logarithm Equations
Solve the following: (Check your solutions!)
1.
3
log (4 7) 2 x − =
2.
2
log (4 7) 3 x − =
3. ln(5 2 ) 2 x − =
4.
3 3
log log ( 2) 1 x x + + =
5.
4 4
log log ( 12) 3 x x + − =
6.
3 3
log log ( 24) 4 x x + − =
7.
4 4
log ( 3) log ( 3) 2 x x + + − =
8.
5 5
log (4 15) log 2 x x + + =
9.
4 4
log ( 2) log ( 1) 1 x x + − − =
10.
2 2
log (4 10) log ( 1) 3 x x + − + =
11.
4 4
log (3 1) log ( 1) 1 x x − − + =
12. log(3 2) log( 1) 1 x x + − − =
13. log(2 1) log( 3) 1 x x − − − =
14. log ( ) log ( )
7 7
1 5 1 x x + + − =
15.
6 6 6
log 3 log 4 log 24 x + =
16.
2 2 2
log ( 5) log log 4 x x + − =
17.
8 8 8
log ( 1) log log 4 x x + − =
18.
1
log( 4) log(3 10) log x x
x
− − − =
19.
3 3 3
log log ( 6) log 27 x x + + =
20.
3 3 3
log ( 9) log ( 6) log 126 x x + + − =
21.
7 7 7
log log (3 11) log 4 x x + − =
22.
6 6 6
1 1
log log 9 log 27
2 3
x = +
23. ln( 2) ln( 4) ln3 x x + = − +
24. 2log log5 log1000 x − =
25. log log( 7) 3log2 x x + + =
Page 436
Logarithm Equations-Answers
1. 4 x =
2.
15
4
x =
3.
2
5
1.195
2
e
x
−
= ≈ −
4. 1 x =
5. 16 x =
6. 27 x =
7. 5 x =
8.
5
4
x =
9. 2 x =
10.
1
2
x =
11. No Solution
12.
12
7
x =
13.
29
8
x =
14. 6 x =
15. 2 x =
16.
5
3
x =
17.
1
3
x =
18. 5 x =
19. 3 x =
20. 12 x =
21. 4 x =
22. 9 x =
23. 7 x =
24. 50 2 x =
25. 1 x =
Page 437
Natural Logarithm Equations
Solve each equation.
1. ln( ) 4 x
2. ln( ) 3 x
3. ln( 5) 3 x
4. ln( 7) 2 x
5. ln( 4) 3 x
6. ln( 2) 2 x
7. ln(3 2) 3 x
8. ln(4 1) 5 x
9. 5ln(2 ) 20 x
10. 6ln(2 ) 37 x
11. 6 2ln( ) 5 x
12. 11 3ln( 1) 6 x
13. ln( 3) 1 x
14. 3 ln( 4) 7 x
15. ln( ) ln(4 1) 2 x x
16. ln( 5) ln( ) 2 x x
17. ln( 5) ln( 3) 2 x x
18. ln( 1) ln( 1) 3 x x
19. ln( 2) ln( 5) 3 x x
20. ln( 2) ln( 1) 1 x x
21. 2ln( 4) ln( ) 2 x e
22.
2
3ln( 1) 5 ln( ) x e
23. ln( 6) ln( 4) ln( ) 2 x x x
24. ln( 3) ln( ) ln( 2) 2 x x x
25. ln( 4) ln( ) ln(4) x x
26. ln(5 1) ln(2 3) ln(2) x x
27. 2ln( ) ln(16) x
28. 2ln( ) ln(33) x
29. 3ln( ) ln(2) x
30. 3ln( ) ln(62) x
Page 438
Natural Logarithm Equations-Answers
1.
4
54.598 e ~
2.
3
20.086 e ~
3.
3
5 15.086 e ÷ ~
4.
2
7 14.389 e + ~
5.
( 3)
4 4.050 e
÷
+ ~
6.
( 2)
2 1.865 e
÷
÷ ~ ÷
7.
3
2
6.029
3
e ÷
~
8.
( 5)
1
0.252
4
e
÷
+
~
9.
4
27.300
2
e
~
10.
37
6
238.297
2
e
| |
|
\ .
~
11.
1
2
0.607 e
| |
÷
|
\ .
~
12.
5
3
1 4.294 e
| |
|
\ .
÷ ~
13.
2
3 4.389 e ÷ ~
14.
8
4 2984.958 e + ~
15.
1 1 16
0.959
8
e + +
~
16.
4 16 4
0.592
2
e ÷ + +
~
17.
2
2 64 16
7.750
2
e + +
~
18.
3
1 4.592 e + ~
19.
3
3
2 5
5.367
1
e
e
÷ ÷
~
÷
20.
2
2.746
1
e
e
÷ ÷
~
÷
21.
3
2
4 0.482 e
| |
|
\ .
÷ ~
22. 1 3.718 e + ~
23.
( ) ( )
2
2 2
10 10 96
18.674
2
e e + + + +
~
24.
( ) ( )
2
2 2 2
3 3 8
9.985
2
e e e + + + +
~
25.
4
1.333
3
~
26. 7
27. 4
28. 33 5.745 ~
29.
3
2 1.260 ~
30.
3
62 3.958 ~
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Section 6.1—Linear Systems in Two Variables
1. 23 ( , )
3. 0 4 ( , )
5. ∅
7.
2 1
3 3
, x x
⎛ ⎞
−
⎜ ⎟
⎝ ⎠
9. 6 2 ( , ) − −
11. 7 1 ( , ) −
13. ∅
15.
3
4
2
,
⎛ ⎞
−
⎜ ⎟
⎝ ⎠
17.
( )
2 3 , x x − +
19. 6 1 ( , ) −
21.
3
0
4
,
⎛ ⎞
−
⎜ ⎟
⎝ ⎠
23.
13 17
5 5
,
⎛ ⎞
−
⎜ ⎟
⎝ ⎠
25. 2 4 ( , ) − −
27. ∅
29.
13 4
14 7
,
− ⎛ ⎞
−
⎜ ⎟
⎝ ⎠
31.
( )
3 5 , x x −
33.
7 5
4 4
,
⎛ ⎞
−
⎜ ⎟
⎝ ⎠
35. ∅
37. 1 4 ( , ) − −
39. 2 5 ( , ) y y +
41. 2 4 ( , ) −
43. ∅
45.
11 15
16 8
,
⎛ ⎞
⎜ ⎟
⎝ ⎠
47.
( )
3 4 , x x − +
49. 6 4 ( , ) − −
51.
2 1
3 4
,
− ⎛ ⎞
⎜ ⎟
⎝ ⎠
53. 2 6 ( , ) −
55.
59 36
22 11
,
⎛ ⎞
− −
⎜ ⎟
⎝ ⎠
57.
1
1
3
,
⎛ ⎞
⎜ ⎟
⎝ ⎠
59. 13 ( , ) −
61.
1 1
23 16
,
⎛ ⎞
⎜ ⎟
⎝ ⎠
Page 466
Answers
Page 467
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Page 490
Section 6.3—Solving Linear Systems using Matrices
1. 3 x 3
3. 2 x 3
5. 6
7. 2
9. 3 −
11. e
13. π
15.
3
4
17. 3 1 ( , ) − −
19.
3 5
2 3
,
− ⎛ ⎞
⎜ ⎟
⎝ ⎠
21. 7 ( , ) y y −
23. ∅
25. 134 ( , , )
27. 3 15 ( , , ) − −
29. 2 4 3 ( , , ) z z z − − +
31. 4 31 ( , , ) −
33. 35 2 ( , , ) − −
35. 0 23 ( , , )
37.
3 2 4
5 9
x y
x y
− =
+ =
39.
5 3 8
2 4 6
8 4
x y z
x y z
x z
− + =
− − =
− + =
41. 2 3 ( , ) − −
43. ∅
45.
( )
15 , −
47.
( )
3 2, y y − +
49.
1
21
2
( , , ) − −
51.
3 5 1
2 2 2
, ,
− ⎛ ⎞
⎜ ⎟
⎝ ⎠
53. ∅
55.
( )
5 5 2 1 , , z z z − + +
57.
( )
2 2 4 , , −
59. 113 ( , , )
61.
( )
5 3 , − −
63.
3 4
5 5
,
− ⎛ ⎞
⎜ ⎟
⎝ ⎠
65.
3
2
,
y
y
+ ⎛ ⎞
⎜ ⎟
⎝ ⎠
67. ∅
69.
( )
31 2 , ,−
71.
( )
2 13 , , −
73.
( )
250 , , −
75.
1 3
1
2 2
, ,
⎛ ⎞
−
⎜ ⎟
⎝ ⎠
77.
( )
5 12 1 , , z z z + −
79. ∅
81.
( )
321 , ,
83.
( )
23 21 , , , −
85. small 35 = ¢, medium 45 = ¢, large 60 = ¢
87. 200 gal-special blend, 100 gal-deep freeze,
300 gal-Lite
Page 491
Answers
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Page 505
Section 6.7—Non-Linear Systems of Equations
1.
( ) ( ) { }
0 2 20 , , ,
3.
( )
6 8
2 0
5 5
, , ,
¦ ¹ | |
÷
´ `
|
\ .
¹ )
5. ( ) { }
2 4 ,
7. No Real Solutions
9. ( ) { }
4 2 ,
11.
( ) ( ) { }
1 4 2 2 , , , ÷ ÷ ÷ ÷
13.
1 1
2 3
2 3
, , ,
¦ ¹ | | | |
÷ ÷
´ `
| |
\ . \ .
¹ )
15.
3 4
4 3
5 5
, , ,
¦ ¹ ÷ | | | |
÷
´ `
| |
\ . \ .
¹ )
17. No Real Solutions
19.
( ) ( ) { }
7 4 5 2 ÷ ÷ , , ,
21.
( )
3 5
0 1
2 4
, , ,
¦ ¹
| |
÷
´ `
|
\ .
¹ )
23.
( ) { }
5 2 ,
25. ( ) ( ) { }
1 1 28 , , , ÷ ÷
27.
( )
4 4
4 4
3 3
, , ,
¦ ¹
| |
÷
´ `
|
\ .
¹ )
29.
( ) ( ) ( ) ( ) { }
2 2 2 2 2 2 2 2 2 2 2 2 , , , , , , , ÷ ÷ ÷ ÷
31.
( ) ( ) ( ) ( ) { }
11 1 1 11 1 1 , , , , , , , ÷ ÷ ÷ ÷
33. No Real Solutions
35.
( ) ( )
1 7 1 7
11 1 1
2 2 2 2
, , , , , , ,
¦ ¹
| | | |
÷ ¦ ¦
÷ ÷ ÷ ÷ ÷
| | ´ `
| |
¦ ¦ \ . \ .
¹ )
37. ( ) { }
01 ,
39.
( ) { }
4 2 2 ,±
41.
( ) ( ) { }
00 11 , , ,
43.
( ) { }
28 ,
45.
( ) ( ) { }
2 2 2 2 , , , ÷
47.
( ) ( ) ( ) ( ) { }
21 2 1 12 1 2 , , , , , , , ÷ ÷ ÷ ÷
49.
( ) ( )
1 1
31 3 1 6 6
2 2
, , , , , , ,
¦ ¹ | | | |
÷ ÷ ÷ ÷
´ `
| |
\ . \ .
¹ )
51.
3
2
2
,
¦ ¹ | |
´ `
|
\ .
¹ )
Page 506
Answers
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