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College Algebra
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Math1314
CollegeAlgebra
Fall2010
LSCNorthHarris
LturnellText BoxLatest UpdateNovember 21, 2012
Materialtakenfrom:
Weltman,Perez,Tiballiunpublishedmaterial
CollegeAlgebraversion 73
byStitzandZeager
GotoLSCNorthHarrisMathDepartmentwebsiteforupdatedandcorrected
versionsofthismaterial.
MathDeptWebsite:nhmath.lonestar.edu
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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 9 1011
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. 1
3x
67. No Solution 69. 1133
,
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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
3-73-2-5
3-43-1-2
3-13
13231
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 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 , ( , )
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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
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. 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. 6
7x =
30. ( ), 31. 32. ( ) ( ), 1 1, 33. 34.
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Section 2.5Quadratic Equations
1. 1,53
3. 1 3,2 2
5. 1,04
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
3i
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 5i
57. 7
0,2
59. 3
2
i
61. 2 5
3
63. { }4 2
65. 4,03
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
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Section 2.7Miscellaneous Equations
1. 2
0, ,35
3. { }0, 5, 2
5. 3,1, 1
2
7. 2 2
2, ,3 3
9. 1, 23
i
11.
1 33,1,
2 2
i
13.
12, ,1 32
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, 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. 80
3
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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 13 33 2 5x x+ =
6. 4 2
3 35 6 0x x + = 7.
1 12 42 1 0x x + =
8. 1 1
2 44 4 0x x + = 9.
1 12 44 9 2 0x x + =
10. 1 1
2 43 2 0x x + = 11.
2 13 32 5 3 0x x =
12. 2 1
3 33 5 2 0x x+ = 13.
4 23 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 12 42 11 2 18x x =
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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 ,16256
x = 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 =
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-2 -53
-43
-1 -23
-13
13
23
1 -1 1 2 3 4 5 6 7
-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. 1
2x 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,
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-2 -1 1 2 3 4 5 6 7 -4 -72-3 -5
2-2 -3
2-1 -1
212
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. 13 2 , ,
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 , ,
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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)
2d M
7. 2 29, (3,3)d M 9. 5 17 2, ,
2 2d M
11. 109 11 1, ,12 8 4
d M 13. 23.05 4.8, ( 0.15, 2)d M 15. 53 13, , 2
2d M
17. 1 12, ,
2x 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
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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
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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
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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.
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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.
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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
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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
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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.
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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 35
( )
( )
( )
( )
( ) ( )
( ) ( )( )
( ) ( )( )
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 1114
2 12 3
3 3 16 6
13 3
( )
( )
( )
( )
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( ) ( )( )
( ) ( )( )( )
f
f
f
f xx
f x fx x
f h fh h
f x h f xh x x h
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21. 3 6
0 0223
2 213
2 42 3 4
3 3 818
4 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 ,
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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 3x+ 5, x > 3
(a) f(4)(b) f(3)
(c) f(3)
(d) f(3.001)
(e) f(3.001)(f) f(2)
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Suggested problems are p. 50: 5-10
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1.5 Function Notation 51
9. Let f(x) =
x2 if x 1
1 x2 if 1 < x 1x 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
46x 2(h) f(x) =
6x 2x2 36
(i) f(x) =3
6x 2x2 + 36
(j) s(t) =t
t 8(k) Q(r) =
r
r 8(l) b() =
8
(m) (y) = 3
y
y 8(n) A(x) =
x 7 +9 x
(o) g(v) =1
4 1v2
(p) u(w) =w 8
5w
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, b2.001c, and bpi + 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.
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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) =4x 13
Domain: (0,)
4. k(x) =
4
x 13
Domain: (0,)
5. (a) 2
(b) 6
(c) 14
(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) 2x3
(g)2
(x 4)3 =2
x3 12x2 + 48x 64
(h)2
x3 4 = 2 4x
3
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
+ 6a 2(h) 3a
2
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
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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)(,12) (12 , 0) (0, 12) (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 well 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, b2.001c = 3, and bpi + 6c = 9
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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 wont look simplified per se, but work through until you can cancel the h.
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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) 2x(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
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Ellen TurnellTypewritten TextANSWERS
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Domain of a Function
Find the domain of the following (write answers in interval notation):
1. 22( )5 6xf x
x x= + +
2. 2( ) 9xf 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, )
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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
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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
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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. 23 1
( ) 1
x if xf x
x 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
+
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
( )( )
3 1 5
3 3 5
1 3
39. Intervals of Increasing: , ( , )Intervals of Decreasing: , ( , )Intervals of Constant: ( , )
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. 23 1
( ) 1
x if xf x
x if x= >
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LturnellText BoxAnswers
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 1
1 1
if xf x x if x
if x
< =
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 from4 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.
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Ellen TurnellText BoxAdditional problems taken from Stitz and Zeager
Suggested assignment: p. 73: 1-10
Ellen TurnellRectangle
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) = 3x (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
54321
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?
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Ellen TurnellText BoxAdditional problems taken from Stitz and Zeager
Suggested assignment: p. 77: 3 (a-j)
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1.7 Graphs of FunctionsStitz 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
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3.6Additional 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( )
4f x x
15. ( ) 3f x x 16. ( ) 1f x x 17. ( ) 4f x x 18. ( ) 1f x x 19. 1( ) 3 3
2f x x 20. 1( ) 2 1
2f 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 1
2f 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 2
2f 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 1
2f 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
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-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
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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
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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
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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
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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
4321
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
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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 31
1
2
3
(a) g(x) = f(x) + 3
(b) h(x) = f(x) 12(c) j(x) = f
(x 23
)(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
) 34. 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
1110987654321 1 2 3 4 5 6 7 8
54321
1
2
3
4
5
y = 3x
x
y
1110987654321 1 2 3 4 5 6 7 8
54321
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 = 3x. With the help of your classmates, find the equivalent vertical scaling produced
by the horizontal scalings y = (2x)3, y = |5x|, y = 327x and y = (12x)2. What abouty = (2x)3, y = | 5x|, y = 327x and y = (12x)2?
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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 12 1 2 3 4
4321
1
2
3
4
(b) y = f(x+ 1)
x
y
(3, 0)
(1, 4)
(1, 0)
(3,2)
4 3 12 1 2 3 4
4321
1
2
3
4
(c) y = 12f(x)
x
y
(2, 0)
(0, 2)
(2, 0) (4,1)4 3 1 1 3 4
4321
1
2
3
4
(d) y = f(2x)
x
y
(1, 0)
(0, 4)
(1, 0)
(2,2)
4 3 2 2 3 4
432
1
2
3
4
(e) y = f(x)
x
y
(2, 0)
(0,4)
(2, 0)
(4, 2)
4 3 12 1 2 3 4
4321
1
2
3
4
(f) y = f(x)
x
y
(2, 0)
(0, 4)
(2, 0)
(4,2)
4 3 1 1 3 4
4321
1
2
3
4
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108 Relations and Functions
(g) y = f(x+ 1) 1
x
y
(3,1)
(1, 3)
(1,1)
(3,3)
4 3 12 1 2 3 4
4321
1
2
3
4
(h) y = 1 f(x)
x
y
(2, 1)
(0,3)
(2, 1)
(4, 3)
4 3 12 1 2 3 4
4321
1
2
3
4
(i) y = 12f(x+ 1) 1
x
y
(3,1)
(1, 1)
(1,1)
(3,2)
4 3 12 1 2 3 4
4321
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
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Ellen TurnellRectangle
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 31
1
2
3
3. (a) g(x) = f(x) + 3
(3, 3)
(0, 6)
(3, 3)
x
y
3 2 1 1 2 31
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 31
1
2
3
(c) j(x) = f(x 23
)
( 7
3, 0)
(23, 3)
(113, 0)x
y
3 2 1 1 2 31
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 21
1
2
(f) c(x) = 35f(x)
(3, 0)
(0, 9
5
)
(3, 0)x
y
3 2 1 1 2 31
1
2
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Ellen TurnellRectangle
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 41
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 21
1
2
3
4
5
6
7
(l) q(x) = 12f(x+42
) 3 = 12f ( 12x+ 2) 3(10,3)
(4, 9
2
)(2,3)
x
y
10987654321 1 2
4321
4. g(x) = 2 3x+ 3 1 or g(x) = 2 3x 3 1
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Ellen TurnellRectangle
Piecewise Functions
Graph the following:
1. = =2 0( )1 0
x if xf x
if x
2. 3 0( )4 0x if x
f xif x
=
Piecewise Functions
1. 2 0
( )1 0
x if xf x
if x= = 2.
3 0( )4 0x if x
f xif x
=
10. 2 1 11 1 x if x
f xx if x
11. 1 -1
0 1 11 1
x if xf x if x
x if x
12. 1 0
1 0x if x
f xif x
13. 1 32 8 3
x if xf x
x if x 14.
10 1
2 1
x if xf x if x
x if x
15.
2
2 4 14 1
1 1
x if xf 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
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y = 0
y = -3
Section 3.7-Linear Functions 1. 3. 5. 7. 9.
11. 54
m =
13. 29
m =
15. m undefined= 17. 2m = 19. 1m =
21. 23. 25.
y = 12x + 3
y = 3x x = 3
f(x) = 2x 4
y = 13x 14
3 y = 2x + 7
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LturnellText BoxAnswers
27. 29.
1 143 3
= y x 31. 2 7= +y x 33. 3= y 35. 4= x 37.
5 294 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.
123
m
b
==
57. 4 1= +y x 59.
3 95 5= +y x
61. 4 83 3
= +y x 63. 3= x 65. 5=y 67.
1 214 4= +y x
69. 5 193 3
= +y x
71. 3 314 4= y x
73. 4=y 75. 3=x
x = 4
y = 2x + 7y =
35x 2
y = 4
x = 2
y = 12x + 3
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Section 3.8Circles 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
4x + 1
22 +
y 532 = 9
4
-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
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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 = 12
C=(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 25
4 4y 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 = 7C=( -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
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Circles-Answers
1. ( )2 25 4 122x y + + = ; Center = 5 ,42
; r=2 3
2. ( )2 23 1 204x y + + = ; Center = 3 , 14
; r=2 5
3. 2 21 3 25
2 2x y + + = ; Center =
1 3,2 2
; r=5
4. ( ) 22 73 162x y + + = ; Center =
73,2
; r=4
5. ( ) 22 12 242x 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. ( ) 22 3 134 2 4x y
+ =
18. ( ) + + = 2
23 27712 4
x y
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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 fD 63. ( )(3)f gD 64. ( )(7)f gD 65. ( )( 5)g f D 66. ( )(3)g fD Additional problems : A. ( )(0)f gD B. ( )(9)g fD C. ( )( 10)f g D D. ( )( 1)f g D E. ( )(3)g fD F. ( )(6)g fD
x
y
g(x)
x
y
f(x)
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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 2 22 42 1 2 7 2 4 6 12
3( )( ) ; ( )( ) ; ( )( ) ; ( )f xf g x x x f g x x x fg x x x x x
g 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 x
gx x x x x x x
9. 2
2 2 3 2 202 15 25 6 15 100 45
+ + = + = = + = = + ( )( ) ; ( )( ) ; ( )( ) ; ( )f x xf g x x x f g x x fg x x x x x xg x
11. 2 2 2 24 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 13 3
( )( ) ; ( )( )f g x g f xx 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
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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) = 3x 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) = 1x 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)
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62 Relations and Functions
1.6.2 Answers
1. (a) i. (f + g)(4) = 16 ii. (gh)(7) = 1187
iii. (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) =
1x
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
xx
Domain: (0,)
2. (a) i. (f + g)(4) = 23 ii. (g h)(1) = 6 iii. (fh)(0) = 32
iv.
(h
g
)(1) = 1
3
(b) i. (f g)(x) = 2x2 + 3x+ 3x+ 1Domain: [0,)
ii. (gh)(x) = 6x 3Domain: (, 2) (2,)
iii.
(f
g
)(x) =
3x 1
2x2 3x 2Domain: [0, 2) (2,)
iv.
(f
h
)(x) = xx+ 13x+ 2
x 23
Domain: [0, 2) (2,)
3. (a) i. (f + g)(3) = 23
ii. (g h)(8) = 1114
iii.
(f
g
)(4) =
22
20
iv. (fh)(8) =
46
4
v. (g + h)(4) = 1618
vi.
(h
g
)(12) = 1
1728
(b) i. (f + g)(x) = x2 36 +6x 2
Domain:
[1
3,)
ii. (g h)(x) = x2 36 1x 4
Domain: (, 4) (4,)iii.
(f
g
)(x) =
6x 2x2 36
Domain:
[1
3, 6
) (6,)
iv. (fh)(x) =
6x 2x 4
Domain:
[1
3, 4
) (4,)
v. (g + h)(x) = x2 36 + 1x 4
Domain: (, 4) (4,)vi.
(h
g
)(x) =
1
(x 4) (x2 36)Domain:(,6) (6, 4) (4, 6) (6,)
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Section 3.10Inverse 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 4f x x = +
51. 1 3 4( )5
xf 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 xx
+= 69.
( ) 4000900
C xx =
-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
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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: 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) 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): 0,2x = 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 )
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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 3f 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 = 81
4
55. x-int(s): 32
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): 0,9x , Maximum value = 405
4
63. 21( ) 1 24
f x x 65. 8c
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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 1f x x 2. 2( ) 3 1f x x 3. 21( ) 1 22f x x 4. 21( ) 3 12f x x 5. 2( ) 2 3f x x 6. 2( ) 2 2 1f x x 7. 2( ) 1 4f x x
8. 2( ) 2 2f x x 9. 2( ) 2 5f x x x 10. 2( ) 2 8f x x x 11. 2( ) 4 7f x x x 12. 2( ) 2 3f x x x 13. 2( ) 6 3f x x x 14. 2( ) 2 4 3f x x x 15. 2( ) 2 2f x x x
Quadratic Functions Worksheet--Answers 1. x-int(s): 3, 5x Axis: 4x ; 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, 4x Axis: 3x ; 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: 1x ; Range: [2, )
4. x-int(s): NONE Axis: 3x ; Range: [1, )
5. x-int(s): 2 3x Axis: 2x ; 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): 12
2x
Axis: 2x ; 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, 3x Axis: 1x ; 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 2x Axis: 2x ; 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 6x Axis: 1x ; Range: [ 6, )
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 )
Quadratic Functions Worksheet--Answers
10. x-int(s): none 2( ) ( 1) 7f x x
Axis: 1x ; 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 6x
2( ) ( 2) 11f x x Axis: 2x ; Range: [ 11, )
-6 -5 -4 -3 -2 -1 1 2 3
-12-11-10-9-8-7-6-5-4-3-2-1
12
Vertex( -2 , -11 )
12. x-int(s): 1, 3x 2( ) ( 1) 4f x x
Axis: 1x ; Range: ( , 4]
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5Vertex( 1 , 4 )
13. x-int(s): 3 6x
2( ) ( 3) 6f x x Axis: 3x ; 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): 2 102
x 2( ) ( 1) 5f x x
Axis: 1x ; 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) 1f x x Axis: 1x ; 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 )
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LturnellStamp
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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
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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
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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
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Section 4.3-Division of Polynomials
1. 62 15
xx
+ + 3. 3 6x + 5. 9 8x 7. 2 25 4
3 1x
x+ +
9. 2 59 7 25
x xx
11. 32 63
xx
+ + 13. 3 2x + 15. 2 52 3 4
2 3x x
x+ + +
17. 3 2 52 3 12 6
x x xx
+ + +
19. 2 22 12 4
3xx x
x x + + +
21. 3 2 1x x x+ + + 23. 2
42 12 3
xx x
+ +
25. 2 23 56 22 4
xx xx+ + +
27. 4 3 2 222 2 3
3 1x x x x
x + + +
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 4
34
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 2
2x x x
x + +
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LturnellText Box #44.This should say 2 and -1 are zeros
Section 4.4The 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. 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. 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 33
2 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
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LturnellText BoxFor an alternate method for 29-48 please see page 337
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Section 4.5Real 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 5 101 2 5 10
3 3 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 1 2 3 ( )( ) ( )x x x 39. 16
2 ,x
41. 1 5 2 ,x 43. 21 2 1
3 , ,x
45. 3 5 2 ,x 47. 1 5 11
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
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Ellen TurnellText Boxf(x) = x3 - 10x - 12
Ellen TurnellText Boxf(x) = 4x3 - 11x - 7
Ellen TurnellText Boxf(x) = 2x4 + 3x3 + x2 - 20x - 20
LturnellText BoxFor an alternate method for 1-20, please see page 337
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Exercises 4.5Graphical 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
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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( ) 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 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
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Section 4.6Complex 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 1
2= ,x
13. 5 55 1 24
= , ,ix 15. 3 1 1 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 5 552 2 5 10 2 4 4 4 4 + + = + + + ( ) ( )
i ix x x x x x x x
25. ( )2 2 21 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.