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Math 1314
College Algebra
Problems and Answers
Fall 2010
LSC – North Harris
Material taken from:
Weltman, Perez, Tiballi unpublished material
“College Algebra” version 73
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 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-12 -10 -8 -6 -4 -2 2 x-1 1 2 3 4 5 6 7 8 9 1011
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x-1 -23
-13
13
23
1
x2 73
83
3 103
113
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 22
= = −x x
13. 82 or 3−
= − =x x 15. No solution 17. 10 14 or 3 3
= = −x x
19. 88 or 3
= = −x x 21. 52
= −x 23. 14 8 or 5 5
= =x x
25. 4 2 or 15 3
= = −x x 27. 5 or 4= = −x x 29. 102 or 3
= − = −x x
31. 13
=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. 112
= −x
43. 5 1 or 3 5
= =x x 45. 113 or 7
= = −x x
51. ( )5 5− , 53. ( )1 1−∞ − ∞, ( , )∪ 55. 4 2−[ , ] 57. 5 9−[ , ] 59. ( 11 1−∞ ∞⎤⎦∪, [ , ) 61. ( 5 7−∞ ∞⎤⎦, [ , )∪
63. ( )5 4− , 65. 13
=x
67. No Solution 69. 1133
⎛ ⎞⎜ ⎟⎝ ⎠
,
Page 15
x-2 -1 1 2 3 4 x-2 -32
-1 -12
12
1 32
2
x-4 -2 2 42/3
x1 2 3 4 5 6 7 8 9 1011x-3-8
3-73
-2-53
-43
-1-23
-13
13
23
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
71. 1 33
⎛ ⎞−∞ − ∞⎜ ⎟⎝ ⎠
, ( , )∪ 73. 1 1 1 or 2 2 2
⎛ ⎞ ⎛ ⎞≠ − −∞ − − ∞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
, ,∪x
75. ( )−∞ ∞, 77. ( 223
, ,⎡ ⎞−∞ − ∞⎤⎦ ⎟⎢⎣ ⎠∪
79. ( )4 8, 81. 7 13−⎛ ⎞
⎜ ⎟⎝ ⎠
,
83. ( 1 5, [ , )−∞ − ∞⎤⎦∪ 85. 9 74 4
, ,⎛ ⎤ ⎡ ⎞−∞ − ∞⎜ ⎟⎥ ⎢⎝ ⎦ ⎣ ⎠∪
87. ( )9 1− −, 89. ( )1 5− ,
91. 10 23 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. 3x ≤
2. 5x <
3. 2 7x + ≤
4. 3 5 10x − <
5. 2 43
x −<
6. 2 3 10x + ≤
7. 4 2 3 9x + − <
8. 3 1 7 10x − + ≤
9. 1 4 7 2x− − < −
10. 1 2 4 1x− − ≤ −
11. 4x ≥
12. 7x >
13. 1 8x + ≥
14. 2 1 7x − ≥
15. 1 35
x −>
16. 3 4 8x + ≥
17. 2 10 16x + ≥
18. 3 1 2 4x − − >
19. 9 2 2 1x− − ≥ −
20. 2 4 5 8x + − >
21. 2 3 12x − <
22. 2 5 9x + ≥
23. 2 3 5 13x − + ≤
24. 4 1 7 13x − − >
25. 2 1 12 5x + + =
26. 4 5 4x− − =
27. 2 5 6x− − ≤
28. 1 02x+ ≥
29. 6 7 0x+ ≤
30. 362 >++x
31. 486 ≤++ x
32. 1 0x + >
33. 5513 <++x
34. 5974 <+− x
Page 17
Absolute Value Inequalities- Answers
1. [ ]3,3−
2. ( )5,5−
3. [ ]9,5−
4. 5 ,53
−
5. ( )10,14−
6. 13 7,2 2
−
7. ( )5,1−
8. [ ]0,2
9. 31,2
−
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. 5x =
27. ( ),−∞ ∞
28. ( ),−∞ ∞
29. 67
x = −
30. ( ),−∞ ∞
31. ∅
32. ( ) ( ), 1 1,−∞ − − ∞∪
33. ∅
34. ∅
Page 18
Page 31
Page 32
Page 33
Section 2.5—Quadratic Equations
1. 1 ,53
3. 1 3,2 2
5. 1 ,04
7. 2,2
9. 1,4
11. 4,5
13. 32
15. 7,3
17. 13
19. 3 55
21. 112
23. 3 7
25. 2,1
27. 3 62
29. 2 2
31. 13
i
33. 2 2
35. 6, 2
37. 3 10
39. 5 3i
41. 5 132
43. 2 32
45. 5 574
47. 1 596i
49. 1 3
51. 4 i
53. 52
55. 3 5i
57. 70,2
59. 32
i
61. 2 53
63. 4 2
65. 4 ,03
67. 1 32
69. 1 53
i
71. 3 77
73. 1 23
i
75. 1, 32
77. 12
i
Page 34
Page 44
Page 45
Section 2.7—Miscellaneous Equations
1. 20, ,35−⎧ ⎫
⎨ ⎬⎩ ⎭
3. { }0, 5, 2− −
5. 3 ,1, 12−⎧ ⎫−⎨ ⎬
⎩ ⎭
7. 2 22, ,3 3−⎧ ⎫
⎨ ⎬⎩ ⎭
9. 1 , 23
i⎧ ⎫±⎨ ⎬⎩ ⎭
11. 1 33,1,
2 2i⎧ ⎫−⎪ ⎪− ±⎨ ⎬
⎪ ⎪⎩ ⎭
13. 12, ,1 32
i⎧ ⎫− ±⎨ ⎬⎩ ⎭
15. 52−⎧ ⎫
⎨ ⎬⎩ ⎭
17. ∅
19. { }4
21. { }9
23. { }1−
25. { }6
27. { }1−
29. { }32
31. { }3
33. { }1−
35. { }3
37. 1 , 23
⎧ ⎫± ±⎨ ⎬⎩ ⎭
39. 3 ,2
i⎧ ⎫± ±⎨ ⎬⎩ ⎭
41. { }2 13± ±
43. 7 32
⎧ ⎫− ±⎪ ⎪±⎨ ⎬⎪ ⎪⎩ ⎭
45. 1 ,827
⎧ ⎫−⎨ ⎬⎩ ⎭
47. 27 1,8 8
⎧ ⎫− −⎨ ⎬⎩ ⎭
49. 1 ,44
⎧ ⎫⎨ ⎬⎩ ⎭
51. { }9
53. 2 , 33
⎧ ⎫−⎨ ⎬⎩ ⎭
55. 1 ,24
⎧ ⎫⎨ ⎬⎩ ⎭
57. 9 15,4 4
⎧ ⎫⎨ ⎬⎩ ⎭
59. 1 3,3 5
⎧ ⎫−⎨ ⎬⎩ ⎭
61. { }7,3−
63. 1 ,22
⎧ ⎫⎨ ⎬⎩ ⎭
65. { }1,15−
67. { }16,11−
69. { }31,33−
71. 803
⎧ ⎫⎨ ⎬⎩ ⎭
Page 46
Quadratic Types of Equations
Find all solutions of the equation. 1. 4 213 40 0x x− + =
2. 4 25 4 0x x− + =
3. 6 32 3 0x x− − =
4. 6 37 8 0x x+ − =
5. 2 1
3 33 2 5x x+ =
6. 4 2
3 35 6 0x x− + =
7. 1 1
2 42 1 0x x− + =
8. 1 1
2 44 4 0x x− + =
9. 1 1
2 44 9 2 0x x− + =
10. 1 1
2 43 2 0x x− + =
11. 2 1
3 32 5 3 0x x− − =
12. 2 1
3 33 5 2 0x x+ − =
13. 4 2
3 34 65 16 0x x− + =
14. 2 110 24 0x x− −− − =
15. 2 13 7 6 0x x− −− − =
16. 2 12 7 4 0x x− −− − =
17. 2 17 19 6x x− −+ =
18. 2 15 43 18x x− −− =
19. 2 16 2x x− −+ =
20. 4 29 35 4 0x x− −− − =
21. ( ) ( )22 7 2 12 0x x+ + + + =
22. ( ) ( )22 5 2 5 6 0x x+ − + − =
23. ( ) ( )23 4 6 3 4 9 0x x+ − + + =
24. ( ) ( )22 2 20 0x x− + − − =
25. ( ) ( )22 1 5 1 3x x+ − + =
26. ( ) ( )1 1
2 42 11 2 18x x− = − −
Page 47
Answers
Quadratic Types of Equations 1. 2 2, 5x = ± ±
2. 2, 1x = ± ±
3. 3 3, 1x = −
4. 2,1x = −
5. 125 ,127
x = −
6. 2 2, 3 3 x = ± ±
7. 1x =
8. 16x =
9. 1 ,16
256x =
10. 16,1x =
11. 1 ,278
x = −
12. 1 , 827
x = −
13. 1 , 648
x = ± ±
14. 2 5,3 8
x = −
15. 3 1,2 3
x = −
16. 12,4
x = −
17. 7 1,2 3
x = −
18. 5 1,2 9
x = −
19. 32,2
x = −
20. 13 ,2
x i= ± ±
21. 5, 6x = − −
22. 71,2
x = − −
23. 13
x = −
24. 7, 2x = −
25. 32,2
x = −
26. 6563,18x =
Page 48
Page 60
Page 61
-2 -53
-43
-1 -23
-13
13
23
1 -1 1 2 3 4 5 6 7
-5 -92
-4 -72
-3 -52
-2 -32
-1 -12
12
1-2 -1 1 2 3 4 5 6
-1 -12
12
12 – 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 -12
12
1 32
2 52
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. 123
, 3. 4 5 , ( , )
5. 9 02
, ( , ) 7. 1 5 ,
9. 12
x 11. 2 3 2 3 ,
13. 4 2 4 2 , , 15. ,
17. No Solution 19. 5 3 2 , ( , ) 21. 1 2 3 3 , ( , ) ( , ) 23. 1 ,
25. 1 302 2
, , 27. 5 3 3 , ,
29. 3 , 31. 2 5,
Page 62
-2 -1 1 2 3 4 5 6 7 -4 -72
-3 -52
-2 -32
-1 -12
12
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. 132
, ,
37. 6 5 , 39. 6 , 41. 2 4 , 43. 2 4 , ( , ) 45. 10 4 , ( , ) 47. 6 1 4 , ( , )
49. 4 2 , ( , ) 51. 13 4 12
, ,
Page 63
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
123456
A
B
C
D
3. 10, (3, 1)d M
5. 317, ( ,1)2
d M
7. 2 29, (3,3)d M
9. 5 17 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. 53 13, , 22
d M
17. 11
2, ,2
x hd h M y
19. (10,11)B
21. 11( ,6)2
23. 10 6 5, 20P A 27. (7, 2);( 9,2) 29. ( 3,12);( 3,2) 31. (0,9);(0, 7) 37. ( 3,0); 37Center radius
Page 87
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
123456
y = x2
-4 -3 -2 -1 1 2 3 4
-3-2-1
123456
y = x4 y = 3 – |x|
28. 29. 30.
x = y2 + 2 y = 2x y = –3x
31. 32. 33.
x2 + y2 = 4 4x2 + 9y2 = 36 y = 1x
Page 99
34. 35. 36.
1x2 + 1
x = |y| + 3
y = 4 – x
37. 38. 39.
x2 – y2 = 1 y2 – x2 = 1
x = y2 – 4
40. 41. 42.
y = 4 – x2 y = x2 + 2
x = 4 – y2
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
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-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
1234
13. 15. 17.
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-6 -5 -4 -3 -2 -1 1 2
-4-3-2-1
1234
Page 104
19. 21. 23.
-4 -3 -2 -1 1 2 3 4 5 6
-2-1
123456
-6 -5 -4 -3 -2 -1 1 2 3 4
-2-1
123456
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
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
123456
49. 51. 53.
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6-2-1
12345678
Page 105
55. 57. 59.
-2 -1 1 2 3 4 5 6-2-1
12345678
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2-1 1 2 3 4 5 6 7 8 910
-4-3-2-1
1234
61. 63. 65.
-2 -1 1 2 3 4 5 6 7 8
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
67. 69. 71.
-4 -3 -2 -1 1 2 3 4
-2-1
123456
73.
75. 77.79.
Page 106
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 50 42 10
1 3 12 3
23 3 3
3
( )( )( )( )( ) ( )
( ) ( )
( ) ( )
ffff x xf x f
xf h f
hf x h f x
h
11.
2
3 130 52 3
1 2 4 32 2 4
23 3 2 12
4 2
( )( )( )( )( ) ( )
( ) ( )
( ) ( )
ffff x x xf x f x
xf h f h
hf x h f x x h
h
13.
2
3 100 72 5
1 2 102 6
23 3 2
2 4
( )( )( )( )( ) ( )
( ) ( )
( ) ( )
ffff x x xf x f x
xf h f h
hf x h f x x h
h
15.
2
3 150 92 5
1 3 5 72 3 7
23 3 3 17
6 3 1
( )( )( )( )( ) ( )
( ) ( )
( ) ( )
ffff x x xf x f x
xf h f h
hf x h f x x h
h
17.
533
0522
511
2 52 2
3 3 53 3
5
( )
( )
( )
( )
( ) ( )
( ) ( )( )
( ) ( )( )
f
f undefined
f
f xx
f x fx x
f h fh h
f x h f xh x x h
19.
136
103
2 111
42 1
2 33 3 1
6 61
3 3
( )
( )
( )
( )
( ) ( )
( ) ( )( )
( ) ( )( )( )
f
f
f
f xx
f x fx x
f h fh h
f x h f xh x x h
Page 119
21. 3 6
0 0223
2 213
2 42 3 4
3 3 81
84 4
( )( )
( )
( )
( ) ( )( )
( ) ( )( )
( ) ( )( )( )
ff
f
xf xx
f x fx x
f h fh h
f x h f xh x x h
23. 2 2 , ( , )
25. ,
27. 52
,
29. 2 5 ( , ] [ , )
31. ,
33. 2 ,
35. ,
37. 3 3 4 4 , ( , ) ( , )
39. ,
41. ,
43. 0 3 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 inthe order given: (1) square root; (2) subtract 13; (3) make the quantity the denominator ofa 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 inthe order given: (1) subtract 13; (2) square root; (3) make the quantity the denominator ofa 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 inthe order given: (1) square root; (2) make the quantity the denominator of a fraction withnumerator 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 inthe 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) = x2 − 3x+ 2, find and simplify the following:
(a) f(3)
(b) f(−1)
(c) f(
32
)(d) f(4x)
(e) 4f(x)
(f) f(−x)
(g) f(x− 4)
(h) f(x)− 4
(i) f(x2)
6. Repeat Exercise 5 above for f(x) =2
x3
7. Let f(x) = 3x2 + 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(
2a
)(h) f(a)
2
(i) f(a+ h)
8. Let f(x) =
x+ 5, x ≤ −3√
9− x2, −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
1.5 Function Notation 51
9. Let f(x) =
x2 if x ≤ −1√
1− x2 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) = x4 − 13x3 + 56x2 − 19
(b) f(x) = x2 + 4
(c) f(x) =x+ 4
x2 − 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
x2 − 36
(i) f(x) =3√
6x− 2
x2 + 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
v2
(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 Thegreatest integer of x, bxc, is defined to be the largest integer k with k ≤ x.
(a) Find b0.785c, b117c, b−2.001c, and bπ + 6c(b) Discuss with your classmates how bxc may be described as a piece-wise defined function.
HINT: There are infinitely many pieces!
(c) Is ba+ bc = bac+ bbc always true? What if a or b is an integer? Test some values, makea conjecture, and explain your result.
8The 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− 13Domain: [0, 169) ∪ (169,∞)
2. g(x) =4√
x− 13Domain: (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) 16x2 − 12x+ 2
(e) 4x2 − 12x+ 8
(f) x2 + 3x+ 2
(g) x2 − 11x+ 30
(h) x2 − 3x− 2
(i) x4 − 3x2 + 2
6. (a)2
27(b) −2
(c)16
27
(d)1
32x3
(e)8
x3
(f) − 2
x3
(g)2
(x− 4)3=
2
x3 − 12x2 + 48x− 64
(h)2
x3− 4 =
2− 4x3
x3
(i)2
x6
7. (a) 16
(b) 4
(c) 12a2 + 6a− 2
(d) 6a2 + 6a− 4
(e) 3a2 + 15a+ 16
(f) 3a2 + 3a+ 14
(g) 12a2 + 6
a − 2
(h) 3a2
2 + 3a2 − 1
(i) 3a2 + 6ah+ 3h2 + 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)[
13 ,∞
)(e)
(13 ,∞
)(f) (−∞,∞)
(g)[
13 , 3)∪ (3,∞)
(h)[
13 , 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
)∪(
12 ,∞
)(p) [0, 25) ∪ (25,∞)
11. The applied domain of P is [0,∞). The range is some subset of the natural numbers becausewe cannot have fractional Sasquatch. This was a bit of a trick question and we’ll address thenotion of mathematical modeling more thoroughly in later chapters. P (0) = 0 means thatthere were no Sasquatch in Portage County in 1803. P (205) ≈ 139.77 would mean there were139 or 140 Sasquatch in Portage County in 2008.
12. (a) b0.785c = 0, b117c = 117, b−2.001c = −3, and bπ + 6c = 9
Page 124
1.6 Function Arithmetic 61
4. Find and simplify the difference quotientf(x+ h)− f(x)
hfor the following functions.
(a) f(x) = 2x− 5
(b) f(x) = −3x+ 5
(c) f(x) = 6
(d) f(x) = 3x2 − x(e) f(x) = −x2 + 2x− 1
(f) f(x) = x3 + 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 6= 0
(l) f(x) = ax2 + bx+ c where a 6= 0
3Rationalize the numerator. It won’t look ‘simplified’ per se, but work through until you can cancel the ‘h’.
Page 125
1.6 Function Arithmetic 63
4. (a) 2
(b) −3
(c) 0
(d) 6x+ 3h− 1
(e) −2x− h+ 2
(f) 3x2 + 3xh+ h2
(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
Domain of a Function
Find the domain of the following (write answers in interval notation):
1. 22( )5 6xf x
x x=
+ +
2. 2( )9
xf xx
=−
3. 23 7( )
6 27xf x
x x+
=− −
4. 3 28( )
8 2 3xf x
x x x+
=− −
5. 24( )
25xf x
x=
−
6. ( ) 3 5f x x= −
7. ( ) 5f x x= +
8. ( ) 3 7f x x= −
9. ( ) 12 24f x x= −
10. ( ) 9 27f x x= − +
11. 5( ) 2f x x= −
12. 1( )3 1
f xx
=+
13. 12( )5
xf xx
=−
14. 2
5 7( )9
xf xx+
=−
15. 2( ) 4f x x= −
16. 2( ) 12 11 5f x x x= + −
17. 2( ) 5 6f x x x= + +
18. 43)( 2 −−= xxxf
19. 2( ) 2 8f x x x= − −
20. 2( ) 9f x x= −
21. 2( ) 100f x x= −
22. 4)( 2 += xxf
23. 2( ) 6 12f x x x= − −
24. 2( ) 15 4 3f x x x= − −
25. 25( )
121xf x
x−
=−
26. 4( ) 3 15f 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 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
12345
-4 -3 -2 -1 1 2 3 4
-3-2-1
123456
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
24. 25. 26.
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
27. 28. 29.
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
Page 141
30. 31. 32.
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-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
12345
-4 -3 -2 -1 1 2 3 4
-2-1
12345
36. 37. 38.
-4 -3 -2 -1 1 2 3 4
-2-1
12345
-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 xf xx if x+ ≤⎧
= ⎨− >⎩
42. 3 0( )
2 0x if xf x
x if x− − ≤⎧
= ⎨+ >⎩
43. 1 2( )
3 9 2x if xf x
x if x+ ≤⎧
= ⎨− + >⎩
44. 3 2( )3 5 2
x if xf x
x if x− ≤⎧
= ⎨− >⎩
45. 2
3 1( )
1x if x
f xx if x
≤⎧⎪= ⎨>⎪⎩
46. 4 2( )2 2
if xf xx if x
≤ −⎧= ⎨
− > −⎩
47. 3 5 2( )
1 2x if xf x
if x+ ≤ −⎧
= ⎨− > −⎩
48. 4 1( )
1 1x if xf x
if x+ ≠⎧
= ⎨− =⎩
49. 2 1 2( )5 2
x if xf xif x
− ≠⎧= ⎨
=⎩
50. 2 0( )
3 0x if xf x
if x⎧ ≠⎪= ⎨
=⎪⎩
51. 2 1( )
4 1x if xf x
if x⎧ ≠
= ⎨=⎩
52.
1 5 22
( ) 3 7 2 31 3
x if x
f x x if xx if x
⎧ + < −⎪⎪
= − + − ≤ ≤⎨⎪ + >⎪⎩
53.
6 41( ) 7 4 22
5 2
x if x
f x x if x
x if x
+ < −⎧⎪⎪= + − ≤ ≤⎨⎪− + >⎪⎩
54. 2 5 0
( ) 5 0 42 4
x if xf x if x
x if x
− + <⎧⎪= ≤ <⎨⎪ − ≥⎩
55. 1 1
( ) 1 11 1
if xf x x if x
if x
− < −⎧⎪= − ≤ <⎨⎪ ≥⎩
56. 1 2
( ) 3 2 32 9 3
x if xf x if x
x if x
− + ≤ −⎧⎪= − < <⎨⎪− + ≥⎩
57. 2 4 1
( ) 2 1 24 2
x if xf x if x
x if x
+ ≤ −⎧⎪= − < <⎨⎪− + ≥⎩
-6 -4 -2 2 4 6
-4
-2
2
4
6F(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
= −∞ ∞
= − ∞
,[ , )
DR
23. ( )0
= −∞ ∞
= ∞
,[ , )
DR
25. ( )( )
= −∞ ∞
= −∞ ∞
,,
DR
27. ( )4
= −∞ ∞
= − ∞
,[ , )
DR
29. 3 31 2
= −= −
[ , ][ , ]
DR
31. ( ){ }All integers
= −∞ ∞
=
,DR
33. ( )
1 0
= −∞ ∞
= − ∞⎡ ⎤⎣ ⎦∪,
[ , )DR
35. ( )
2 2
= −∞ ∞
= −∞ ∞∪,
( , ) ( , )DR
37. 0 40 2
==
[ , ][ , ]
DR
39. ( )( )
( )
Intervals of Increasing: 4 1
Intervals of Decreasing: 2
Intervals of Constant: 4 1 2
− −
∞
−∞ − −∪
,,, ( , )
41. 2 1 0( )5 0
x if xf x
x if x+ ≤⎧
= ⎨− >⎩
43. 1 2( )
3 9 2x if x
f xx if x+ ≤⎧
= ⎨− + >⎩
45. 2
3 1( )
1x if x
f xx if x
≤⎧= ⎨
>⎩
Page 144
47. 3 5 2( )
1 2x if x
f xif x
+ ≤ −⎧= ⎨
− > −⎩ 49.
2 1 2( )5 2
x if xf x
if x− ≠⎧
= ⎨=⎩
51. 2 1( )
4 1x if xf x
if x⎧ ≠
= ⎨=⎩
53.
6 41( ) 7 4 22
5 2
x if x
f x x if x
x if x
+ < −⎧⎪⎪= + − ≤ ≤⎨⎪− + >⎪⎩
55. 1 1
( ) 1 11 1
if xf x x if x
if x
− < −⎧⎪= − ≤ <⎨⎪ ≥⎩
57. 2 4 1
( ) 2 1 24 2
x if xf 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 theequation 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 to3, 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 thepoint which corresponds to x = 2. Since the point (2, 0) is on the graph, we have f(2) = 0.
Page 146
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 anyintercepts and test for symmetry.
(a) f(x) =x− 2
3(b) f(x) =
√5− x (c) f(x) = 3
√x (d) f(x) =
1
x2 + 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
x3
(d) f(x) = 4
(e) f(x) = 0
(f) f(x) = x6 − x4 + x2 + 9
(g) f(x) = −x5 − x3 + x
(h) f(x) = x4+x3+x2+x+1
(i) f(x) =√
5− x
(j) f(x) = x2 − 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 theequation 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
1.7 Graphs of Functions—Stitz and Zeager Book ANSWERS p. 73:1-15 1. 4 4[ , ]− 2. 3 3[ , ]− 3. 2 0( )f = 4. 2 0 2 0( , ),( , )− 5. 0 3( , ) 6. 2 2,x = − 7. 4 2 2 4[ , ] ( , ]− − ∪ 8. 2 solutions 9. 4 0[ , )− 10. 0 4( , ] 11. 0 3( , ) 12. none 13. 3 14. 3− 15. yes, even p. 77: 3 (a-j) ANSWERS
Page 148
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) 3f x x 2. 2( ) ( 1) 4f x x 3. ( ) 1 2f x x 4. ( ) 2 1f x x
5. 3( ) 3 2f x x 6. 3( ) 2 2 1f x x
7. ( ) 4 4f x x 8. ( ) 2 3 3f x x
9. 3( ) 2 3f x x 10. 3( ) 1 2f x x 11. ( ) 2 1 1f x x 12. 2( ) 2 3 2f x x
13. 2( ) 3f x x 14. 21( )4
f x x
15. ( ) 3f x x 16. ( ) 1f x x
17. ( ) 4f x x 18. ( ) 1f x x
19. 1( ) 3 32
f x x 20. 1( ) 2 12
f x x
21. ( ) 1 3f x x 22. 1( ) 1 42
f x x
23. 3( ) 2 1 2f x x 24. ( ) 3f x x
25. 31( ) 2 12
f x x 26. 3( ) 3f x x
27. 2( ) 2 3 5f x x 28. 31( ) 1 42
f x x
29. 3( ) 2 1 2f x x 30. 3( ) 2 3f x x
31. 1( ) 3 22
f x x 32. ( ) 2 1 1f x x
33. 31( ) 1 32
f x x 34. 1( ) 4 22
f x x
35. 3( ) 2 1f x x 36. 31( ) 4 12
f x x
37. 3( ) 4 1f x x 38. 3( ) 3 1f x x
39. ( ) 2 3 1f x x 40. 3( ) 5 3f x x
Page 162
Page 163
Page 164
Page 165
Page 166
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
Section 3.6-Graphing Techniques 1. 3. 5.
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
-2 -1 1 2 3 4 5 6
-5-4-3-2-1
12345
7. 9. 11. 13. 15. 17. 19. 21. 23.
-1 1 2 3 4 5 6 7-2-1
12345678
-1 1 2 3 4 5 6 7-2-1
12345678
-6 -5 -4 -3 -2 -1 1 2
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-3-2-1
1234567
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
Page 167
25. 27. 29. 31. 33. 35. 37. 39. 41i.
-8 -7 -6 -5 -4 -3 -2 -1 1 2
-4-3-2-1
1234
-3 -2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-6-5-4-3-2-1 1 2 3 4 5 6
-6-5-4-3-2-1
123456
41ii. 41iii. 41iv.
-6-5-4-3-2-1 1 2 3 4 5 6
-6-5-4-3-2-1
123456
-6-5-4-3-2-1 1 2 3 4 5 6
-6-5-4-3-2-1
123456
-6-5-4-3-2-1 1 2 3 4 5 6
-6-5-4-3-2-1
123456
-4 -3 -2 -1 1 2 3
-5-4-3-2-1
1234
-4 -3 -2 -1 1 2 3
-5-4-3-2-1
1234
-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
1234
-4 -3 -2 -1 1 2 3-2-1
1234567
-6 -5 -4 -3 -2 -1 1 2
-4-3-2-1
1234
Page 168
41v. 41vi. 41vii.
-6-5-4-3-2-1 1 2 3 4 5 6
-6-5-4-3-2-1
123456
-6-5-4-3-2-1 1 2 3 4 5 6
-6-5-4-3-2-1
123456
-6-5-4-3-2-1 1 2 3 4 5 6
-5-4-3-2-1
12345
41viii. 43i. 43ii.
-9-8-7-6-5-4-3-2-1 1 2 3
-9-8-7-6-5-4-3-2-11234
-4 -3 -2 -1 1 2 3 4 5 6 7 8
-4-3-2-1
123456
-4 -3 -2 -1 1 2 3 4 5 6 7 8
-4-3-2-1
123456
43iii. 43iv. 43v.
-4 -3 -2 -1 1 2 3 4 5 6 7 8
-4-3-2-1
123456
-4 -3 -2 -1 1 2 3 4 5 6 7 8
-6-5-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4 5 6 7 8
-4-3-2-1
123456
Page 169
43vi. 43vii. 43viii.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-4-3-2-1
123456
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6-5-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4 5 6 7 8
-4-3-2-1
123456
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
12345
-4 -3 -2 -1 1 2 3 4
-2-1
12345
-2
-1
1
2
73. 75. 77.
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-5-4-3-2-1
12345
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
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 = 12f(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 = 12f(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 = 12S(−x+ 1)
(d) y = 12S(−x+ 1) + 1
Page 171
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)− 12
(c) j(x) = f(x− 2
3
)(d) a(x) = f(x+ 4)
(e) b(x) = f(x+ 1)− 1
(f) c(x) = 35f(x)
(g) d(x) = −2f(x)
(h) k(x) = f(
23x)
(i) m(x) = −14f(3x)
(j) n(x) = 4f(x− 3)− 6
(k) p(x) = 4 + f(1− 2x)
(l) q(x) = −12f(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 youranswer 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 sameas 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 =(
12x)2
. What about
y = (−2x)3, y = | − 5x|, y = 3√−27x and y =
(−1
2x)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 = 12f(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 = 12f(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 = 12S(−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 = 12S(−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)− 12
(−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
)
(23, 3
)
(113, 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) = 35f(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(
23x)
(− 9
2, 0
)
(0, 3)
(92, 0
)x
y
−4 −3 −2 −1 1 2 3 4
−1
1
2
3
(i) m(x) = −14f(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)
(12, 7
)
(2, 4)
x
y
−1 1 2
−1
1
2
3
4
5
6
7
(l) q(x) = − 12f(x+42
)− 3 = − 1
2f(12x+ 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 0x if x
f xif x
2. 3 0
( )4 0x if x
f xif x
≥= − <
3. 2 3 1
( )3 2 1
x if xf x
x if x− + <
= − ≥
4. 2
1 0( )
0x if x
f xx if x+ <
= ≥
5. 3
2 0( ) 3 0
0
x if xf x if x
x if x
− ≤ <
= − = >
6. 3 3 0
( ) 2 00
x if xf x if x
x if x
+ − ≤ <
= = >
7. 3
3
1 0( )0
x if xf xx if x
+ >= − ≤
8. ( )0
2 1 0x if x
f xx if x
<= + ≥
9. ( )4 22 2 2− <
= − ≥
x if xf x
x if x
10. ( ) 2
1 11 1
− + ≤= − >
x if xf x
x if x
11. ( )1 -1
0 1 11 1
− − <= − ≤ ≤ − >
x if xf x if x
x if x
12. ( )1 01 0− ≤
= − >
x if xf x
if x
13. ( )1 32 8 3− ≤
= − + >
x if xf x
x if x
14. ( )1
0 12 1
<= =− + >
x if xf x if x
x if x
15. ( )2
2 4 14 11 1
− + <= = + >
x if xf x if x
x if x
16. ( ) 01 0x if x
f xif x
≠=
=
17. ( ) 1 12 1 − ≠
= =
x if xf x
if x
Page 177
Piecewise Functions
1. 2 0
( )1 0
x if xf x
if x
≠⎧= ⎨ =⎩
2. 3 0
( )4 0x if x
f xif x
≥⎧= ⎨− <⎩
3. 2 3 1
( )3 2 1
x if xf x
x if x− + <⎧
= ⎨ − ≥⎩
4. 2
1 0( )
0x if x
f xx if x+ <⎧
= ⎨≥⎩
5.
⎧ − ≤ <⎪
= − =⎨⎪ >⎩
3
2 0( ) 3 0
0
x if xf x if x
x if x 6.
3 3 0( ) 2 0
0
x if xf x if x
x if x
⎧ + − ≤ <⎪
= =⎨⎪ >⎩
7. ⎧ + >⎪= ⎨− ≤⎪⎩
3
3
1 0( )0
x if xf xx if x
8. ( )0
2 1 0x if x
f xx if x
<⎧= ⎨ + ≥⎩
9. ( )4 22 2 2− <⎧
= ⎨ − ≥⎩
x if xf x
x if x
Page 178
10. ( ) 2
1 11 1
− + ≤⎧= ⎨ − >⎩
x if xf x
x if x1 11. ( )
1 -10 1 1
1 1
x if x
f x if x
x if x
− − <⎧⎪= − ≤ ≤⎨⎪ − >⎩
12. ( )1 0
1 0x if x
f xif x
− ≤⎧= ⎨− >⎩
13. ( )1 3
2 8 3− ≤⎧
= ⎨− + >⎩
x if xf x
x if x 14. ( )
10 1
2 1
x if x
f x if x
x if x
<⎧⎪= =⎨⎪− + >⎩
15. ( )2
2 4 14 1
1 1
x if x
f x if x
x if x
− + <⎧⎪= =⎨⎪ + >⎩
16. ( ) 01 0x if x
f xif x
⎧ ≠= ⎨
=⎩ 17. ( ) 1 1
2 1⎧ − ≠
= ⎨=⎩
x if xf x
if x
Page 179
Page 196
Page 197
Page 198
Page 199
y = 0
y = -3
Section 3.7-Linear Functions 1. 3. 5. 7. 9.
11. 5
4m −
=
13. 29
m =
15. m undefined= 17. 2m = 19. 1m = −
21. 23. 25.
y =⎝⎜⎛12⎠⎟⎞x + 3
y = 3x x = 3
f(x) = 2x – 4
y =⎝⎜⎛13⎠⎟⎞x – 14
3 y = –2x + 7
Page 200
27. 29.
1 143 3
= −y x
31. 2 7= − +y x 33. 3= −y 35. 4= −x
37. 5 29
4 4−
= +y x
39. 2 19 9
= −y x
41. 4=x 43. 2=y
45. 2 25
= −y x
47. 2
7mb
= −=
49. 352
m
b
=
= − 51.
04
mb
==
53. m undefinedb none
==
55. 12
3
m
b
=
=
57. 4 1= +y x
59. 3 9
5 5−
= +y x
61. 4 83 3
= +y x
63. 3= −x 65. 5=y
67. 1 21
4 4−
= +y x
69. 5 193 3
= +y x
71. 3 31
4 4−
= −y x
73. 4=y 75. 3=x
x = –4
y = –2x + 7y =
⎝⎜⎛35⎠⎟⎞x – 2
y = 4
x = –2
y =⎝⎜⎛12⎠⎟⎞x + 3
Page 201
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
1234
x2 + y2 = 1
-5 -4 -3 -2 -1 1 2 3 4 5 6
-6-5-4-3-2-1
1234
(x – 1)2 + (y + 1)2 = 16
-1 1 2 3 4 5 6 7 8 9 10
-4-3-2-1
123456
(x – 5)2 + (y – 2)2 = 9
-5 -4 -3 -2 -1 1 2 3 4 5 6-2-1
12345678
(x – 1)2 + (y – 3)2 = 20
-7 -6 -5 -4 -3 -2 -1 1 2
-7-6-5-4-3-2-1
12(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
= 94
-10 -8 -6 -4 -2 2 4 6
-4-2
2468
10(x + 3)2 + (y – 2)2 = 25
-7 -6 -5 -4 -3 -2 -1 1 2
-7-6-5-4-3-2-1
12
(x + 2)2 + (y + 5)2 = 5
-10 -8 -6 -4 -2 2 4 6-2
2468
1012
(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
12345678r = 1
2C=(1/2,5/2)
-8 -7 -6 -5 -4 -3 -2 -1 1 2
-4-3-2-1
123456r = 70
2C=(-3,1)
39. No graph 41. 2 2( 2) ( 5) 25x y+ + − = 43. 2 2( 6) ( 6) 36x y− + + =
45. 3 254 4
y x−
= +
-2 2 4 6
-8-7-6-5-4-3-2-1
1
r = 2C=( 5 , -3 )
-6 -4 -2 2
-6
-5
-4
-3
-2
-1
1r = 7 C=( -2 , -3 )
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
123456789
101112
r = 13 C=( 0 , 7 )
-4 -2 2 4
-6-5-4-3-2-1
12r = 6 C=( 0 , -2 )
-4 -2 2 4 6 8 10
-4
-2
2
4
6
8
10r = 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 2 415 8 04
x y x y+ + − + =
2. 2 2 3 2952 02 16
x y x y+ − + − =
3. 2 2 453 02
x y x y+ − + − =
4. 2 2 216 7 04
x y x y+ − + + =
5. 2 2 794 04
x y x y+ + − − =
6. 2 2 279 3 02
x y x y+ − − + =
7. 2 2 1 2 2567 02 3 144
x y x y+ + + − =
8. 2 2 10 35 03 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
25 4 122
x y + + − =
; Center = 5 ,42
−
; r=2 3
2. ( )2
23 1 204
x y − + + =
; Center = 3 , 14
−
; r=2 5
3. 2 21 3 25
2 2x y − + + =
; Center = 1 3,2 2
−
; r=5
4. ( )2
2 73 162
x y − + + =
; Center = 73,2
−
; r=4
5. ( )2
2 12 242
x y + + − =
; Center = 12,2
−
; r=2 6
6. 2 29 3 9
2 2x y − + − =
; Center = 9 3,2 2
; r=3
7. 2 21 1 18
4 3x y + + + =
; Center = 1 1,4 3
− −
; r=3 2
8. 2 25 1 4
3 2x y + + − =
; Center = 5 1,3 2
−
; r=2
9. ( ) ( )2 28 3 9x y− + + =
10. ( ) ( )+ + − =2 24 5 25x y
11. ( ) ( )2 28 3 64x y− + + =
12. ( ) ( )2 24 5 16x y+ + − =
13. 2 2 34x y+ = 14. 2 2 53x y+ =
15. ( ) ( )2 22 3 13x y− + − =
16. ( ) ( )2 23 1 32x y− + + =
17. ( )2
2 3 1342 4
x y − + − =
18. ( ) − + + =
223 2771
2 4x y
Page 217
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 − 62. ( )(1)g f 63. ( )(3)f g 64. ( )(7)f g 65. ( )( 5)g f − 66. ( )(3)g f Additional problems : A. ( )(0)f g B. ( )(9)g f C. ( )( 10)f g − D. ( )( 1)f g − E. ( )(3)g f F. ( )(6)g f
x
y
g(x)
x
y
f(x)
Page 233
Section 3.9-Operations on Functions 1. 2 3 15 2 4 6 7 3
2 3( )( ) ; ( )( ) ; ( )( ) ; ( )f xf g x x f g x x fg x x x x
g x⎛ ⎞ +
+ = − − = + = − − =⎜ ⎟ −⎝ ⎠
3. 2 2 3 22
2 42 1 2 7 2 4 6 123
( )( ) ; ( )( ) ; ( )( ) ; ( )f xf g x x x f g x x x fg x x x x xg x
⎛ ⎞ −+ = + − − = − + − = − + − =⎜ ⎟ +⎝ ⎠
5. 2
2 2 3 2 5 26 4 4 8 28 126
( )( ) ; ( )( ) ; ( )( ) ; ( )f x xf g x x x f g x x x fg x x x x xg x
⎛ ⎞ + ++ = + − − = + + = − − − =⎜ ⎟ −⎝ ⎠
7. ( )( ) ( )( ) ( )( ) ( )
3 1 7 2 2 43 2 3 2 3 2 3
( )( ) ; ( )( ) ; ( )( ) ; ( )x x f xf g x f g x fg x xgx x x x x x x
⎛ ⎞+ + ++ = − = = =⎜ ⎟− + − + − + −⎝ ⎠
9. 2
2 2 3 2 202 15 25 6 15 1005
( )( ) ; ( )( ) ; ( )( ) ; ( )f x xf g x x x f g x x fg x x x x xg x
⎛ ⎞ + −+ = + − − = − = + − − =⎜ ⎟ +⎝ ⎠
11. 2 2 224 4 4
4( )( ) ; ( )( ) ; ( )( ) ( ); ( )
( )f xf g x x x f g x x x fg x x x xg x
⎛ ⎞+ = + − − = − + = − =⎜ ⎟ −⎝ ⎠
13. 2 218 48 39 6 17( )( ) ; ( )( )f g x x x g f x x= − + = +
15. 2 210 24 4 4( )( ) ; ( )( )f g x x x g f x x x= − + = + −
17. 22 2( )( ) ; ( )( )f g x x g f x x= − = −
19. 3 1 1
3( )( ) ; ( )( )xf g x g f x
x x+
= =+
21. 3 3 4 82 2 1
( )( ) ; ( )( )x xf g x g f xx x+ −
= =− + +
23. 1 1
3 3( )( ) ; ( )( )f g x g f x
x x= =
− −
25. 9 4x + 27. 32− 29. ( ),−∞ ∞
31. 2− 33. 236 42 16x x− + 35. 76 37. 3− 47. 22 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
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.(gh
)(x)
vi. (h− f)(x)
2. Let f(x) = 3√x− 1, g(x) = 2x2 − 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) = x2 − 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
62 Relations and Functions
1.6.2 Answers
1. (a) i. (f + g)(4) = 16 ii. (g−h)(7) =118
7iii. (fh)(25) =
1
5iv.
(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.(gh
)(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
2iv.
(h
g
)(−1) =
1
3
(b) i. (f − g)(x) = −2x2 + 3x+ 3√x+ 1
Domain: [0,∞)
ii. (gh)(x) = −6x− 3
Domain: (−∞, 2) ∪ (2,∞)
iii.
(f
g
)(x) =
3√x− 1
2x2 − 3x− 2
Domain: [0, 2) ∪ (2,∞)
iv.
(f
h
)(x) = −x
√x+ 1
3x+ 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) = x2 − 36 +√
6x− 2
Domain:
[1
3,∞)
ii. (g − h)(x) = x2 − 36− 1
x− 4
Domain: (−∞, 4) ∪ (4,∞)
iii.
(f
g
)(x) =
√6x− 2
x2 − 36
Domain:
[1
3, 6
)∪ (6,∞)
iv. (fh)(x) =
√6x− 2
x− 4
Domain:
[1
3, 4
)∪ (4,∞)
v. (g + h)(x) = x2 − 36 +1
x− 4
Domain: (−∞, 4) ∪ (4,∞)
vi.
(h
g
)(x) =
1
(x− 4) (x2 − 36)
Domain:(−∞,−6) ∪ (−6, 4) ∪ (4, 6) ∪ (6,∞)
Page 236
Page 250
Page 251
Page 252
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 34( )
5x
f x− +=
53. not one-to-one 55.
1 2( ) 4, 0f x x x− = − ≥
57. 1( ) 2f x x− = +
59. 1( ) 1f x x− = − −
61. not one-to-one 63. not one-to-one 65.
( )31( ) 1f x x− = − 67.
1 2 1( ) xf x
x− +
=
69. ( ) 4000
900C x
x−
=
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
f-1(x) = x + 4
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
f-1(x) = x – 52
-2 -1 1 2 3 4 5 6 7 8-2-1
12345678
f-1(x) = x2 + 4;x ≥ 0
-4 -3 -2 -1 1 2 3 4
-6-5-4-3-2-1
123456
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234f-1(x) = x – 2
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234f-1(x) = 3 x – 2
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-3
-2
-1
1
2
3
4
Page 254
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. 3 132 2
,x
19. 3 x 21. 1 3 x 23. Axis: 0x ; Range: ( ,0]
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
Vertex( 0 , 0 )
25. Axis: 0x ; Range: [ 3, )
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
Vertex( 0 , -3 )
27. Axis: 2x ; Range: ( ,0]
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
Vertex( 2 , 0 )
29. Axis: 2x ; Range: [2, )
-4 -3 -2 -1 1 2
-2
-1
1
2
3
4
Vertex( -2 , 2 )
31. Axis: 1
2x ;
Range: ( ,1]
-3 -2 -1 1 2 3
-3
-2
-1
1
2
Vertex(1/2,1)
33. Axis: 5x ; 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 6x 2( ) ( 1) 6f 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) 7f x x
-4 -3 -2 -1 1 2 3 4
-12
-10
-8
-6
-4
-2
2
Vertex( -1 , -7 )
39. x-int(s): 0,2x 2( ) ( 1) 4f x x
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
Vertex( 1 , 4 )
Page 268
41. x-int(s): 4x 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) 1f x x
-6 -5 -4 -3 -2 -1 1 2
-2
-1
1
2
3
Vertex( -4 , 1 )
45. x-int(s): none
21 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): 3 192
x
23 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): 24,
3x
25 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): 0,9x
29 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 = 814
55. x-int(s): 32
x ,
Minimum value = 9
57. x-int(s): 32
x ,
Maximum value = 0 59. x-int(s): none, Maximum value = 2 61. x-int(s): 0,9x ,
Maximum value = 4054
63. 21( ) 1 24
f x x
65. 8c
Page 269
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. 2a = 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
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
1234
-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
123456
-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
1234
-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
12345678
Page 288
Page 297
Page 298
Page 299
Section 4.3-Division of Polynomials
1. 62 15
xx
− ++
3. 3 6x + 5. 9 8x −
7. 2 25 43 1
xx
+ +−
9. 2 59 7 25
x xx
− − −−
11. 32 63
xx
− ++
13. 3 2x +
15. 2 52 3 42 3
x xx
+ + +−
17. 3 2 52 3 12 6
x x xx
− + + −+
19. 22
2 12 43
xx xx x
−− + +
+ −
21. 3 2 1x x x+ + +
23. 2
42 12 3
xx x
+ −− +
25. 22
3 56 22 4
xx xx+
− + +−
27. 4 3 22
22 2 33 1
x x x xx
− + − + −+
29. 312
xx
− +−
31. 172 42
xx
− ++
33. 2 62 43
x xx
− − −−
35. 2 22 32
x xx
− − ++
37. 2 43 14
x xx
− − ++
39. 2 34 2 412
x xx
− − ++
41. 3 22 4 6x x x− − −
43. 3 2 516 12 434
x xx
− + −+
45. 2 5 25x x− + 47. 5 4 3 22 4 8 16 32x x x x x+ + + + + 49. 3x +
51. 3 2 122 2 2 22
x x xx
− + − −+
Page 300
Page 310
Page 311
Page 312
Section 4.4—The Remainder and Factor Theorems 1. 34 3. 316 5. 0
7. 31081
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. No, it is not a zero 27. Yes, it is a zero 29. No, it is not a factor 31. No, it is not a factor 33. Yes, it is a factor 35. Yes, it is a factor 37. 2 3 1 1 ( )( )( )x x x 39. 25 2 3 1 ( ) ( )( )x x x 41. 34 3 1 ( )( )x x 43. 2 1 2 1 ( )( )( )x x x 45. 3 22 29 30 x x x
47. 3 21 332 2
x x x or 3 22 6 3 x x x
49. 4 3 214 123
x x x x or 4 3 23 14 3 36 x x x x
53. 272
k
55. 2 k
Page 313
Page 323
Page 324
Page 325
Section 4.5—Real Zeros of Polynomial Functions
1. (a) 51 32
, ,x ,(b) 143
,x , (c) 0 2 ,x (d) 1 3 x
21. 1 3 9 , ,
23. 1 2 51 2 53 3 3
, , , , ,
25. 1 31 2 3 62 2
, , , , ,
27. 1 1 11 28 2 4
, , ,
29. 5 2 1 ( )( )( )x x x 31. 1 2 1 1 ( )( )( )x x x 33. 22 1 ( ) ( )x x 35. 2 2 2 1 3 1 ( )( )( )x x x x 37. 2 21 2 3 ( ) ( )x x x
39. 162
,x
41. 1 5 2 ,x
43. 21 2 13
, ,x
45. 3 5 2 ,x
47. 1 5 112 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
Page 334
Page 335
Page 336
Section 4.6—Complex Zeros of Polynomial Functions
1. 95
x
3. 1 174
x
5. 1 3 ,x i 7. 0 3 ,x i 9. 2 1 7 ,x
11. 1 2 2 12
,x
13. 5 55 0 24
, ,ix
15. 3 1 1 2 , ,x i
17. 32
,x i
19. 2 3 , ,x i 21. 2 9 1 1 3 3 ( ) ( )( )( )x x x x i x i
23. 2 5 55 5 552 2 5 10 24 4 4 4
( ) ( ) i ix x x x x x x x
25. 2 1 2 3 2 3 ( ) ( )( )( )x x x x i x i
27. 2 25 1 5 5 ( )( )( )( )x x x x x i x i
29. 2 22 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 3 210 38 64 40 x x x x 37. 4 3 214 98 406 841 x x x x 39. 4 28 16 x x
Page 339
Exercises 4.5—Graphical Approach-Alternate Method In exercises 29-47, use the Rational Zeros Theorem, the given graph, and synthetic division to find all zeros of each polynomial function.
29. 3 2( ) 4 7 10f x x x x 31. 3 2( ) 2 2 1f x x x x
33. 3 2( ) 3 4f x x x 35. 4 3 2( ) 6 25 4 4f x x x x x
37. 5 4 3 2( ) 4 8 7 17 3 9f x x x x x x 39. 3 2( ) 2 12 6f x x x x
41. 3 2( ) 4 8f x x x 43. 4 3 2( ) 3 5 7 3 2f 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 10f x x x x 47. 4 3 2( ) 4 7 2 4 1f x x x x x
Exercises 4.6— Graphical Approach-Alternate Method In exercises 9-19, use the Rational Zeros Theorem, the given graph, and synthetic division to find all zeros of each polynomial function.
9. 3 2( ) 10 12f x x x 11. 3( ) 4 11 7f x x x
13. 4 3 2( ) 2 3 20 20f x x x x x 15. 4 2( ) 4 4 12 9f x x x x
17. 4 3 2( ) 4 12 13 12 9f x x x x x 19. 5 4 3 2( ) 3 3 9 4 12f 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
Page 351
Page 352
Page 353
Section 4.8—Rational Functions 1. Rational 3. Not 5. Not 7. Rational
9. 13
x
11. 12
x
13. ,
15. 1 2 ,x 17. Vertical: 3x ; Horizontal: 5y 19. Vertical: 2 x ; Horizontal: none 21. Vertical: none; Horizontal: 0y
23. Vertical: 2 13
,x ; Horizontal: 13
y
25. Vertical: 4 x ; Horizontal:
0y ; Intercepts: 102
,
27. Vertical: 1x ; Horizontal: 3y ; Intercepts: 0 0,
29. Vertical: 2 x ; Horizontal: 0y ; Intercepts: 0 0,
31. Vertical: none; Horizontal: 0y ; Intercepts: 0 0,
33. Vertical: 3 1 ,x ; Horizontal: 2y ; Intercepts:
0 0,
35. Vertical: 0x ; Horizontal: 1y ; 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
37. Vertical: 1x ; Horizontal: 0y ; Intercepts: 0 0,
39. Vertical: none; Horizontal: 1y ; 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
Exercises 4.9—Ellipse and Hyperbola In exercises 1-16, find the vertices and foci for each ellipse. Graph each ellipse.
1. 2 2
116 4
x y
2. 2 2
125 16
x y
3. 2 2
19 36
x y
4. 2 2
116 49
x y
5. 2 2
125 16
x y
6. 2 2
149 36
x y
7. 2 2
149 81
x y
8. 2 2
164 100
x y
9. 2 2
181 25
4 16
x y
10. 2 2
19 25
4 4
x y 11. 2 225 4 100 x y 12. 2 29 4 36 x y
13. 2 24 49 196 x y 14. 2 24 25 100 x y
15. 2 22 4 x y 16. 2 28 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
12345
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-6-5-4-3-2-1
12345
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-6-5-4-3-2-1
12345
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-6-5-4-3-2-1
12345
Page 370
In exercises 21-32, find the standard form of the equation of each ellipse satisfying the given information.
21. Vertices: 3 0, ; minor axis of length 2 .
22. Vertices: 0 5, ; minor axis of length 4 .
23. Vertices: 0 4, ; and passing through the point 1 3( , ). 24. Vertices: 3 0, ; and passing through the point 2 1( , ) . 25. Major axis is horizontal with length 8; length of minor axis = 4 ;
center is 0 0( , ) 26. Major axis is horizontal with length 12; length of minor axis =6 ;
center is 0 0( , ) 27. Vertices: 8 0 , ; Foci: 5 0 ,
28. Vertices: 6 0 , ; Foci: 2 0 ,
29. Vertices: 0 7, ; Foci: 0 4,
30. Vertices: 0 4, ; Foci: 0 3,
31. Foci: 2 0 , ; y-intercepts: 3 and 3
32. Foci: 0 2, ; x-intercepts: 2 and 2 In exercises 33-46, find the vertices, asymptotes, and foci for each hyperbola. Graph each hyperbola.
33. 2 2
19 25
x y 34.
2 2
116 25
x y
35. 2 2
1100 64
x y
36. 2 2
1144 81
x y
37. 2 2
125 64
y x
38. 2 2
19 64
y x
39. 2 2
149 16
y x
40. 2 2
1100 5
y x
41. 2 24 36 y x 42. 2 216 64 y x
43. 2 29 4 36 x y 44. 2 24 25 100 x y
45. 2 29 25 225 y x 46. 2 216 9 144 x y
Page 371
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. Vertices: 3 0 , ; Foci: 4 0 ,
52. Vertices: 5 0 , ; Foci: 7 0 ,
53. Vertices: 0 1, ; Foci: 0 3,
54. Vertices: 0 2, ; Foci: 0 6,
55. Foci: 6 0 , ; Endpoints of conjugate axis: 0 3,
56. Foci: 0 4, ; Endpoints of conjugate axis: 2 0 ,
57. Transverse axis is horizontal with length 8; length of conjugate axis = 4 ; center is 0 0( , )
58. Transverse axis is vertical with length 6 ; length of conjugate axis =14; center is 0 0( , )
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-6-5-4-3-2-1
12345
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
12345
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-9-8-7-6-5-4-3-2-1
12345678
Page 372
4.9—Ellipse and Hyperbola 1. Vertices: 4 0 , ; 3. Vertices: 0 6, ; 5. Vertices: 5 0 , ;
Foci: 2 3 0 , Foci: 0 3 3, Foci: 3 0 ,
7. Vertices: 0 9, ; 9. Vertices: 9 02
, ; 11. Vertices: 0 5, ;
Foci: 0 4 2, Foci: 299 04
, Foci: 0 21,
13. Vertices: 7 0 , ; 15. Vertices: 0 2, ; 17. 2
2 14
x y ; Foci: 3 0 ,
Foci: 3 5 0 , Foci: 0 2, 19. 2 2
19 25
x y; Foci: 0 4,
21. 2 2
19 1
x y
23. 2 2
116 16
7
x y
25. 2 2
116 4
x y
-5 -4 -3 -2 -1 1 2 3 4 5
-6-5-4-3-2-1
12345
-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
123456
-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
123456
-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
27. 2 2
164 39
x y
29. 2 2
133 49
x y
31. 2 2
113 9
x y
33. Vertices: 3 0 , ; 35. Vertices: 10 0 , ; 37. Vertices: 0 5, ;
Foci: 34 0 , Foci: 2 41 0 , Foci: 0 89,
Asymptotes: 53
y x Asymptotes: 45
y x Asymptotes: 58
y x
39. Vertices: 0 7, ; 41. Vertices: 0 3, ; 43. Vertices: 2 0 , ;
Foci: 0 65, Foci: 0 3 5, Foci: 13 0 ,
Asymptotes: 74
y x Asymptotes: 12
y x Asymptotes: 32
y x
45. Vertices: 0 5, ; 47. 2 2
19 4
x y; Foci: 13 0 ,
Foci: 0 34, 49. 2 2
116 9
y x
; Foci: 0 5,
Asymptotes: 53
y x 51. 2 2
19 7
x y 55.
2 2
127 9
x y
53. 2 2
11 8
y x 57.
2 2
116 4
x y
-5 -4 -3 -2 -1 1 2 3 4 5
-6-5-4-3-2-1
12345
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
-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
12345678
-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
2468
1012
-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
Exercises 5.1 and 5.2—Exponential Functions In exercises 1-8, graph each function by making a table of coordinates.
1. 5( ) xf x 2. 4( ) xf x 3. 13
( )x
f x
4. 12
( )x
f x 5. 32
( )x
f x 6. 43
( )x
f x
7. 0 6( ) . xf x 8. 0 9( ) . xf x
By translating, reflecting, and stretching the graph of 2( ) xf x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function.
9. 12 ( ) xf x 10. 22 ( ) xf x 11. 2 2 ( ) xf x
12. 2 1 ( ) xf x 13. 22 3 ( ) xf x 14. 12 3 ( ) xf x
15. 2( ) xf x 16. 2 1 ( ) xf x 17. 2 ( ) xf x
18. 12 ( ) xf x 19. 12 ( ) xf x 20. 12 3 ( ) xf x
21. 1 2 32
( ) xf x 22. 11 22
( ) xf x 23. 12 2 1 ( ) xf x
By translating, reflecting, and stretching the graph of 3( ) xf x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function.
24. 23 ( ) xf x 25. 13 ( ) xf x 26. 3 1 ( ) xf x
27. 3 2 ( ) xf x 28. 13 3 ( ) xf x 29. 23 3 ( ) xf x
30. 3( ) xf x 31. 3 1 ( ) xf x 32. 3 ( ) xf x
33. 13 ( ) xf x 34. 13 ( ) xf x 35. 13 2 ( ) xf x
36. 1 3 32
( ) xf x 37. 12 3 ( ) xf x 38. 12 3 1 ( ) xf x
By translating, reflecting, and stretching the graph of ( ) xf x e , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function.
39. 1( ) xf x e 40. 2( ) xf x e 41. 2 ( ) xf x e
42. 1 ( ) xf x e 43. 1 2 ( ) xf x e 44. 2 1 ( ) xf x e
45. ( ) xf x e 46. 2 ( ) xf x e 47. ( ) xf x e
48. 1 ( ) xf x e 49. 2 ( ) xf x e 50. 2 1 ( ) xf x e
51. 2 3 ( ) xf x e 52. 112
( ) xf x e 53. 11 12
( ) xf x e
Page 392
In exercises 54-77, solve each exponential equation in by expressing each
side as a power of the same base and then equating the exponents.
54. 4 8x 55. 13 81x 56. 128
x
57. 3 1 110100
x 58. 25 125x 59. 23 81x
60. 3 54 8x x 61. 4 116 4x 62. 1366
x
63. 1 1255
x 64. 2 121xee
65. 2xe e
66. 1 49 27x 67. 128 4
x x 68. 2 8 3 12 8x x
69. 223 81x 70. 6 4125 0.2x 71. 9 3x x
72. 19 27x 73.
3
46 6x
74.
2
67 7x
75. 142
x 76. 3193
x 77. 421x
xee
In exercises 78-86, solve the following by using the appropriate formulas.
0 1
( )ntrA t A
n or 0( ) rtA t A e
78. If Manhattan Island had been purchased from Native Americans in 1626 for
$24, and if that $24 was immediately deposited in an account that paid
interest at the rate of 5% compounded yearly, what would be the value of
the investment in 1998? What would be the value of the investment if
the interest rate were 6%?
79. Find the value of a $5000 investment for 10 years at an annual interest
rate of 6% if interest is compounded:
a) annually
b) monthly
c) daily
d) each minute
e) continuously
80. Find the value on an investment after 8 years if $3500 is invested at an
annual interest rate of 4.5% compounded quarterly.
Page 393
81. Find the value on an investment after 30 months if $1200 is invested at
an annual interest rate of 5.25% compounded daily.
82. An investor with $2000 to invest for 3 years may invest this money at an
annual rate of 8% compounded monthly or at an annual rate of 6%
compounded continuously. Which strategy is better? How much more money
is made by following the better strategy?
83. Find the value on an investment after 8 years if $1500 is invested at an
annual interest rate of 4% compounded continuously.
84. Find the value on an investment after 5 years if $10,000 is invested at
an annual interest rate of 8% compounded continuously.
85. How much must be invested initially at 5% compounded continuously if the
value of the investment is to be $4,200 after 7 years?
86. Determine the annual interest rate r if an initial investment of $1000
is to grow to $1,250 in 3 years if interest is compounded monthly.
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
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
7. 9. D: ( , ) , R: (0, ) 11. D: ( , ) , R: (0, ) Asymptote: 0y Asymptote: 2y
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
13. D: ( , ) , R: ( 3, ) 15. D: ( , ) , R: (0, ) 17. D: ( , ) , R: ( ,0) Asymptote: 3y Asymptote: 0y Asymptote: 0y
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
Page 395
19. D: ( , ) , R: ( ,0) 21. D: ( , ) , R: ( 3, ) 23. D: ( , ) , R: (1, ) Asymptote: 0y Asymptote: 3y Asymptote: 1y
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-6-5-4-3-2-1
12
-4 -3 -2 -1 1 2 3 4
-2-1
123456
25. D: ( , ) , R: (0, ) 27. D: ( , ) , R: (2, ) 29. D: ( , ) , R: ( 3, ) Asymptote: 0y Asymptote: 2y Asymptote: 3y
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-2-1
123456
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
31. D: ( , ) , R: (1, ) 33. D: ( , ) , R: (0, ) 35. D: ( , ) , R: ( , 2) Asymptote: 1y Asymptote: 0y Asymptote: 2y
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
Page 396
37. D: ( , ) , R: (0, ) 39. D: ( , ) , R: (0, ) 41. D: ( , ) , R: ( 2, ) Asymptote: 0y Asymptote: 0y Asymptote: 2y
-4 -3 -2 -1 1 2 3 4
-2-1
123456
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
43. D: ( , ) , R: (2, ) 45. D: ( , ) , R: (0, ) 47. D: ( , ) , R: ( ,0) Asymptote: 2y Asymptote: 0y Asymptote: 0y
-4 -3 -2 -1 1 2 3 4
-2-1
123456
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
49. D: ( , ) , R: ( ,0) 51. D: ( , ) , R: ( 3, ) 53. D: ( , ) , R: (1, ) Asymptote: 0y Asymptote: 3y Asymptote: 1y
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-6-5-4-3-2-1
12
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
Page 397
55. 3
57. 1
59. 2
61. 38
63. 32
65. 52
67. 4
69. 2
71. 0,4
73. 5
75. 1
4
77. 4
3
79. a) 8954 24$ . b) 9096 98$ . c) 9110 14$ . d) 9110 58$ . e) 9110 60$ . f) 9110 60$ . 80. 5006 58$ . 82. Compounded monthly, 146 04$ . more 84. 14918 25$ . 86. 7 5. %
Page 398
Page 407
Page 408
Page 409
Section 5.3—Logarithms 1. 100 000 5log( , ) =
3. 31 29
log ⎛ ⎞ = −⎜ ⎟⎝ ⎠
5. ( )36162
log =
7. ( )loge P t= or ( )ln P t=
9. 21 1
2 4=
11. 1 11010
− =
13. 0 1e = 15. 3N t= 17. 0 19. 3− 21. 2− 23. 12
25. 12
27. 19
27. 1 857. 29. 223.− 31. 0 861. 33. 1 365. 35. a) 3 27 3log = and 4 16 2log =
b) 6 40 2 0588log .= and 4 16 2log =
37. a) 243x = b) 2x = c) 2 1,x = − 39. a) 0 8 0969log . .= − ; 80 1 9031log .= ; b) 10 times 41. a) 5 1293. b) 1 0995. c) 2 9887.
d) 7 9367.− e) 44453.−
45. 132
(log log ) loga a ax y z+ −
47. 2 4 3 6( log log ) (log log )a a a aw z x+ − +
49. 11 33 4
(log log ) ( log log )a a a ax y z w+ − +
51. 2 12 4 12
(log ( )) log ( )a ax x+ − +
53. 2 1log ( )a x x +
55. 4 3
logay x
z
57. 1 1
log( )( )a
xx x+ −
Page 410
342 Exponential and Logarithmic Functions
6.1.1 Exercises
1. Evaluate the expression.
(a) log3(27)
(b) log6(216)
(c) log2(32)
(d) log6
(136
)(e) log8(4)
(f) log36(216)
(g) log 15(625)
(h) log 16(216)
(i) log36(36)
(j) log(
11000000
)(k) log(0.01)
(l) ln(e3)
(m) log4(8)
(n) log6(1)
(o) log13
(√13)
(p) log36
(4√
36)
(q) 7log7(3)
(r) 36log36(216)
(s) log36
(36216
)(t) ln
(e5)
(u) log(
9√
1011)
(v) log(
3√
105)
(w) ln(
1√e
)(x) log5
(3log3(5)
)(y) log
(eln(100)
)2. Find the domain of the function.
(a) f(x) = ln(x2 + 1)
(b) f(x) = log7(4x+ 8)
(c) f(x) = ln(4x− 20)
(d) f(x) = log(x2 + 9x+ 18
)(e) f(x) = log
(x+ 2
x2 − 1
)(f) f(x) = log
(x2 + 9x+ 18
4x− 20
)(g) f(x) = ln(7− x) + ln(x− 4)
(h) f(x) = ln(4x− 20) + ln(x2 + 9x+ 18
)
(i) f(x) = log(x2 + x+ 1
)(j) f(x) = 4
√log4(x)
(k) f(x) = log9(|x+ 3| − 4)
(l) f(x) = ln(√x− 4− 3)
(m) f(x) =1
3− log5(x)
(n) f(x) =
√−1− x
log 12(x)
(o) f(x) = ln(−2x3 − x2 + 13x− 6)
3. For each function given below, find its inverse from the ‘procedural perspective’ discussed inExample 6.1.5 and graph the function and its inverse on the same set of axes.
(a) f(x) = 3x+2 − 4
(b) f(x) = log4(x− 1)
(c) f(x) = −2−x + 1
(d) f(x) = 5 log(x)− 2
4. Show that logb 1 = 0 and logb b = 1 for every b > 0, b 6= 1.
5. (Crazy bonus question) Without using your calculator, determine which is larger: eπ or πe.
6. (The Logarithmic Scales) There are three widely used measurement scales which involvecommon logarithms: the Richter scale, the decibel scale and the pH scale. The computationsinvolved in all three scales are nearly identical so pay close attention to the subtle differences.
Page 411
6.1 Introduction to Exponential and Logarithmic Functions 345
6.1.2 Answers
1. (a) log3(27) = 3
(b) log6(216) = 3
(c) log2(32) = 5
(d) log6
(136
)= −2
(e) log8(4) = 23
(f) log36(216) = 32
(g) log 15(625) = −4
(h) log 16(216) = −3
(i) log36(36) = 1
(j) log 11000000 = −6
(k) log(0.01) = −2
(l) ln(e3)
= 3
(m) log4(8) = 32
(n) log6(1) = 0
(o) log13
(√13)
= 12
(p) log36
(4√
36)
= 14
(q) 7log7(3) = 3
(r) 36log36(216) = 216
(s) log36
(36216
)= 216
(t) ln(e5) = 5
(u) log(
9√
1011)
= 119
(v) log(
3√
105)
= 53
(w) ln(
1√e
)= −1
2
(x) log5
(3log3 5
)= 1
(y) log(eln(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) ∪(
12 , 2)
3. (a) f(x) = 3x+2 − 4f−1(x) = log3(x+ 4)− 2
x
y
y = f(x) = 3x+2 − 4
y = f−1(x) = log3(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) = log4(x− 1)f−1(x) = 4x + 1
x
y
y = f(x) = log4(x− 1)
y = f−1(x) = 4x + 1
−2−1 1 2 3 4 5 6
−2
−1
1
2
3
4
5
6
Page 412
6.2 Properties of Logarithms 355
6.2.1 Exercises
1. Expand the following using the properties of logarithms and simplify. Assume when necessarythat all quantities represent positive real numbers.
(a) ln(x3y2)
(b) log2
(128
x2 + 4
)(c) log5
( z25
)3
(d) log(1.23× 1037)
(e) ln
(√z
xy
)(f) log5
(x2 − 25
)(g) log√2
(4x3)
(h) log 13(9x(y3 − 8))
(i) log(1000x3y5
)(j) log3
(x2
81y4
)(k) ln
(4
√xy
ez
)(l) log6
(216
x3y
)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) log2(x) + log2(y)− log2(z)
(d) log3(x)− 2 log3(y)
(e) 12 log3(x)− 2 log3(y)− log3(z)
(f) 2 ln(x)− 3 ln(y)− 4 ln(z)
(g) log(x)− 13 log(z) + 1
2 log(y)
(h) −13 ln(x)− 1
3 ln(y) + 13 ln(z)
(i) log2(x) + log 12(x− 1)
(j) log2(x) + log4(x− 1)
(k) log5(x)− 3
(l) log7(x) + log7(x− 3)− 2
(m) ln(x) + 12
3. Use an appropriate change of base formula to convert the following expressions to ones withthe indicated base.
(a) 7x−1 to base e
(b) log3(x+ 2) to base 10(c)
(2
3
)xto base e
(d) log(x2 + 1) to base e
4. Use the appropriate change of base formula to approximate the following logarithms.
(a) log3(12)
(b) log5(80)
(c) log6(72)
(d) log4
(1
10
)(e) log 3
5(1000)
(f) log 23(50)
Page 413
6.2 Properties of Logarithms 357
6.2.2 Answers
1. (a) 3 ln(x) + 2 ln(y)
(b) 7− log2(x2 + 4)
(c) 3 log5(z)− 6
(d) log(1.23) + 37
(e) 12 ln(z)− ln(x)− ln(y)
(f) log5(x− 5) + log5(x+ 5)
(g) 3 log√2(x) + 4
(h) −2+log 13(x)+log 1
3(y−2)+log 1
3(y2+2y+4)
(i) 3 + 3 log(x) + 5 log(y)
(j) 2 log3(x)− 4− 4 log3(y)
(k) 14 ln(x) + 1
4 ln(y)− 14 −
14 ln(z)
(l) 12− 12 log6(x)− 4 log6(y)
(m) 13 ln(x)− ln(10)− 1
2 ln(y)− 12 ln(z)
2. (a) ln(x4y2)
(b) log
(1000
x
)(c) log2
(xyz
)(d) log3
(x
y2
)(e) log3
(√x
y2z
)(f) ln
(x2
y3z4
)
(g) log
(x√y
3√z
)(h) ln
(3
√z
xy
)(i) log2
(x
x− 1
)(j) log2
(x√x− 1
)(k) log5
( x
125
)(l) log7
(x(x− 3)
49
)(m) ln (x
√e)
3. (a) 7x−1 = e(x−1) ln(7)
(b) log3(x+ 2) =log(x+ 2)
log(3)
(c)
(2
3
)x= ex ln( 2
3)
(d) log(x2 + 1) =ln(x2 + 1)
ln(10)
4. (a) log3(12) ≈ 2.26186
(b) log5(80) ≈ 2.72271
(c) log6(72) ≈ 2.38685
(d) log4
(1
10
)≈ −1.66096
(e) log 35(1000) ≈ −13.52273
(f) log 23(50) ≈ −9.64824
Page 414
Exercises 5.4—Logarithmic Functions In exercises 1-8, sketch the graphs of each pair of functions on the same set of axes. Label all asymptotes.
1. 5( ) xf x and 5( ) logg x x
2. 4( ) xf x and 4( ) logg x x
3. 14
( )x
f x and 14
( ) logg x x
4. 12
( )x
f x and 12
( ) logg x x
5. ( ) xf x e and ( ) lng x x
6. 10( ) xf x and ( ) logg x x By translating, reflecting, and stretching the graph of ( ) logf x x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function. 7. 1 ( ) log( )f x x 8. 2 ( ) log( )f x x
9. 1 ( ) logf x x 10. 2 ( ) logf x x
11. 2 3 ( ) log( )f x x 12. 1 4 ( ) log( )f x x
13. ( ) logf x x 14. 2( ) logf x x
15. 1 ( ) logf x x 16. 2 ( ) logf x x
17. 2 1 ( ) log( )f x x 18. 3 2 ( ) log( )f x x By translating, reflecting, and stretching the graph of ( ) lnf x x , obtain the graphs of the following functions. Give the domain, range, and equation of any asymptotes of the function. 19. ( ) ( 1)f x ln x 20. ( ) ( 3)f x ln x
21. 4f ( x ) ln x 22. ( ) 3f x lnx
23. ( ) ( 2) 1f x ln x 24. ( ) ( 2) 4f x ln x
25. ( ) lnf x x 26. 12( ) logf 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
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
-4 -3 -2 -1 1 2 3 4
-4-3-2-1
1234
7. D: (1, ) , R: ( , ) 9. D: (0, ) , R: ( , ) 11. D: ( 2, ) , R: ( , ) Asymptote: 1x Asymptote: 0x Asymptote: 2x
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-6-5-4-3-2-1
12
13. D: (0, ) , R: ( , ) 15. D: (0, ) , R: ( , ) 17. D: (1, ) , R: ( , ) Asymptote: 0x Asymptote: 0x Asymptote: 1x
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-2-1
123456
Page 422
19. D: ( 1, ) , R: ( , ) 21. D: (0, ) , R: ( , ) 23. D: (2, ) , R: ( , ) Asymptote: 1x Asymptote: 0x Asymptote: 2x
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-2-1
123456
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
25. D: (0, ) , R: ( , ) 27. D: ( 2, ) , R: ( , ) Asymptote: 0x Asymptote: 2x
-2 -1 1 2 3 4 5 6
-4-3-2-1
1234
-2 -1 1 2 3 4 5 6
-2-1
123456
Page 423
Page 431
Page 432
Section 5.5—Exponential and Logarithmic Equations 1. a) 1x =
b) 302
,x =
c) 0 3,x =
d) 0x =
3. 1 8.x ≈
5. 0 58.x ≈ −
7. 1 20.x ≈ − or ( )ln P t=
9. 0 29.x ≈
11. 0 and 2 81.x x= ≈
13. 0 18.x ≈ −
15. 0 21.x ≈
17. 1
216x =
19. 1
10x =
21. 3x =
23. 4x =
25. 92
x =
27. 1x = −
29. 4x =
31. 0 5,x =
33. 65
x =
35. 32
x =
37. 1 and 10000x =
39. 10000x =
41. 2x =
Page 433
Exponential Equations
Solve. Round answers to 3 decimal places.
1. 4 21x
2. 7 35x
3. 52 11x
4. 17 20x
5. 13 16x
6. 3 25 7x
7. 29 17 6x
8. 4 1 11xe
9. 9 107xe
10. 53 25xe
11. 2 34 120xe
12. 3 41000 3000xe
13. 4 13 1 19xe
14. 23(2 9 ) 11x
15. 3 4 23xe
16. 1 2 53 7x x
17. 8 2 5 22 3x x
18. 3 1 5 45 7x x
Page 434
Exponential Equations-Answers
1. ln21ln4
x 2.196
2. ln35ln7
x 1.827
3. ln115ln2
x 0.692
4. ln20 1ln7
x 0.540
5. ln16 1ln3
x 1.524
6. ln7 2ln53ln5
x 0.264
7. ln23 2ln9
x 3.427
8. 1 ln114
x 0.349
9. 107ln9
x
2.476
10. 1 25ln5 3
x
0.424
11. 3 ln302
x 3.201
12. 4 ln33
x 1.700
13. 1 ln64
x 0.198
14. 5ln 3
2ln9x 0.116
15. 3 ln234
x 0.034
16. ln3 5ln7ln3 2ln7
x
3.877
17. 2ln3 2ln28ln2 5ln3
x
68.760
18. 4 ln7 ln53ln5 5ln7
x
1.260
Page 435
Logarithm Equations
Solve the following: (Check your solutions!)
1. 3log (4 7) 2x − =
2. 2log (4 7) 3x − =
3. ln(5 2 ) 2x− =
4. 3 3log log ( 2) 1x x+ + =
5. 4 4log log ( 12) 3x x+ − =
6. 3 3log log ( 24) 4x x+ − =
7. 4 4log ( 3) log ( 3) 2x x+ + − =
8. 5 5log (4 15) log 2x x+ + =
9. 4 4log ( 2) log ( 1) 1x x+ − − =
10. 2 2log (4 10) log ( 1) 3x x+ − + =
11. 4 4log (3 1) log ( 1) 1x x− − + =
12. log(3 2) log( 1) 1x x+ − − =
13. log(2 1) log( 3) 1x x− − − =
14. log ( ) log ( )7 71 5 1x x+ + − =
15. 6 6 6log 3 log 4 log 24x + =
16. 2 2 2log ( 5) log log 4x x+ − =
17. 8 8 8log ( 1) log log 4x x+ − =
18. 1log( 4) log(3 10) logx xx
− − − =
19. 3 3 3log log ( 6) log 27x x+ + =
20. 3 3 3log ( 9) log ( 6) log 126x x+ + − =
21. 7 7 7log log (3 11) log 4x x+ − =
22. 6 6 61 1log log 9 log 272 3
x = +
23. ln( 2) ln( 4) ln3x x+ = − +
24. 2log log5 log1000x − =
25. log log( 7) 3log2x x+ + =
Page 436
Logarithm Equations-Answers
1. 4x =
2. 154
x =
3. 25 1.195
2ex −
= ≈ −
4. 1x =
5. 16x =
6. 27x =
7. 5x =
8. 54
x =
9. 2x =
10. 12
x =
11. No Solution
12. 127
x =
13. 298
x =
14. 6x =
15. 2x =
16. 53
x =
17. 13
x =
18. 5x =
19. 3x =
20. 12x =
21. 4x =
22. 9x =
23. 7x =
24. 50 2x =
25. 1x =
Page 437
Page 461
Page 462
Page 463
Section 6.1—Linear Systems in Two Variables 1. 2 3( , )
3. 0 4( , )
5. ∅
7. 2 13 3
,x x⎛ ⎞−⎜ ⎟⎝ ⎠
9. 6 2( , )− −
11. 7 1( , )−
13. ∅
15. 3 42
,⎛ ⎞−⎜ ⎟⎝ ⎠
17. ( )2 3,x x− +
19. 6 1( , )−
21. 3 04
,⎛ ⎞−⎜ ⎟⎝ ⎠
23. 13 175 5
,⎛ ⎞−⎜ ⎟⎝ ⎠
25. 2 4( , )− −
27. ∅
29. 13 414 7
,−⎛ ⎞−⎜ ⎟⎝ ⎠
31. ( )3 5,x x −
33. 7 54 4
,⎛ ⎞−⎜ ⎟⎝ ⎠
35. ∅
37. 1 4( , )− −
39. 2 5( , )y y+
41. 2 4( , )−
43. ∅
45. 11 1516 8
,⎛ ⎞⎜ ⎟⎝ ⎠
47. ( )3 4,x x− +
49. 6 4( , )− −
51. 2 13 4
, −⎛ ⎞⎜ ⎟⎝ ⎠
53. 2 6( , )−
55. 59 3622 11
,⎛ ⎞− −⎜ ⎟⎝ ⎠
57. 1 13
,⎛ ⎞⎜ ⎟⎝ ⎠
59. 1 3( , )−
61. 1 123 16
,⎛ ⎞⎜ ⎟⎝ ⎠
Page 464
Page 483
Page 484
Page 485
Page 486
Page 487
Page 488
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. 34
17. 3 1( , )− −
19. 3 52 3
,−⎛ ⎞⎜ ⎟⎝ ⎠
21. 7( , )y y−
23. ∅
25. 1 3 4( , , )
27. 3 1 5( , , )− −
29. 2 4 3( , , )z z z− − +
31. 4 3 1( , , )−
33. 3 5 2( , , )− −
35. 0 2 3( , , )
37.
3 2 45 9x yx y− =+ =
39.
5 3 82 4 6
8 4
x y zx y zx z
− + =− − =
− + =
41. 2 3( , )− −
43. ∅
45. ( )1 5,−
47. ( )3 2,y y− +
49. 12 12
( , , )− −
51. 3 5 12 2 2
, ,−⎛ ⎞⎜ ⎟⎝ ⎠
53. ∅
55. ( )5 5 2 1, ,z z z− + +
57. ( )2 2 4, ,−
59. 1 1 3( , , )
61. ( )5 3,− −
63. 3 45 5
, −⎛ ⎞⎜ ⎟⎝ ⎠
65. 3
2,y y+⎛ ⎞
⎜ ⎟⎝ ⎠
67. ∅
69. ( )3 1 2, ,−
71. ( )2 1 3, ,−
73. ( )2 5 0, ,−
75. 1 3 12 2
, ,⎛ ⎞−⎜ ⎟⎝ ⎠
77. ( )5 1 2 1, ,z z z+ −
79. ∅
81. ( )3 2 1, ,
83. ( )2 3 2 1, , ,−
85. small 35= ¢, medium 45= ¢, large 60= ¢
87. 200 gal-special blend, 100 gal-deep freeze,
300 gal-Lite
Page 489
Page 502
Page 503
Section 6.7—Non-Linear Systems of Equations 1. ( ) ( ){ }0 2 2 0, , ,
3. ( )6 8 2 05 5
, , ,⎧ ⎫⎛ ⎞ −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
5. ( ){ }2 4,
7. No Real Solutions
9. ( ){ }4 2,
11. ( ) ( ){ }1 4 2 2, , ,− − − −
13. 1 12 32 3
, , ,⎧ ⎫⎛ ⎞ ⎛ ⎞− −⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭
15. 3 44 3
5 5, , ,⎧ ⎫−⎛ ⎞ ⎛ ⎞−⎨ ⎬⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎩ ⎭
17. No Real Solutions
19. ( ) ( ){ }7 4 5 21, , ,− −
21. ( )3 5 0 12 4
, , ,⎧ ⎫⎛ ⎞ −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
23. ( ){ }5 2,
25. ( ) ( ){ }1 1 2 8, , ,− −
27. ( )4 4 4 43 3
, , ,⎧ ⎫⎛ ⎞ −⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
29. ( ) ( ) ( ) ( ){ }2 2 2 2 2 2 2 2 2 2 2 2, , , , , , ,− − − −
31. ( ) ( ) ( ) ( ){ }1 1 1 1 1 1 1 1, , , , , , ,− − − −
33. No Real Solutions
35. ( ) ( )1 7 1 7 1 1 1 12 2 2 2
, , , , , , ,⎧ ⎫⎛ ⎞ ⎛ ⎞−⎪ ⎪− − − − −⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭
37. ( ){ }0 1,
39. ( ){ }4 2 2,±
41. ( ) ( ){ }0 0 1 1, , ,
43. ( ){ }2 8,
45. ( ) ( ){ }2 2 2 2, , ,−
47. ( ) ( ) ( ) ( ){ }2 1 2 1 1 2 1 2, , , , , , ,− − − −
49. ( ) ( ) 1 13 1 3 1 6 62 2
, , , , , , ,⎧ ⎫⎛ ⎞ ⎛ ⎞− − − −⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭
51. 3 22
,⎧ ⎫⎛ ⎞⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
Page 504
