weltman revised (1)

Math1314

CollegeAlgebra

Fall2010

LSCNorthHarris

LturnellText BoxLatest UpdateNovember 21, 2012

Materialtakenfrom:

Weltman,Perez,Tiballiunpublishedmaterial

CollegeAlgebraversion 73

byStitzandZeager

GotoLSCNorthHarrisMathDepartmentwebsiteforupdatedandcorrected

versionsofthismaterial.

MathDeptWebsite:nhmath.lonestar.edu

Ellen TurnellRectangle

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

,

Page 15

LturnellText BoxAnswers

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 , ( , )

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

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.

Page 18

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

Page 34


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

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 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 =

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 ,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 =

Page 48

-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,

Page 62


-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 , ,

Page 63


LturnellRectangle

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

Page 87



LturnellRectangle

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

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

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

( )

( )

( )

( )

( ) ( )

( ) ( )( )

( ) ( )( )( )

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 3 818

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 ,

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 3x+ 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

Ellen TurnellText Box

Ellen TurnellTypewritten TextAdditional Exercises taken from Stitz and Zeager Book

Suggested problems are p. 50: 5-10


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.

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) =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

Page 123



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

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 wont look simplified per se, but work through until you can cancel the h.

Page 125

Ellen TurnellTypewritten TextTaken from Stitz and Zeager


1.6 Function Arithmetic 63

4. (a) 2

(b) 3(c) 0

(d) 6x+ 3h 1(e) 2x h+ 2(f) 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

Page 126

Ellen TurnellTypewritten TextANSWERS


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, )

Page 128

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. 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= >

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 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.

Page 146

Ellen TurnellTypewritten Text

Ellen TurnellText BoxAdditional problems taken from Stitz and Zeager

Suggested assignment: p. 73: 1-10


1.7 Graphs of Functions 77

1.7.2 Exercises

1. Sketch the graphs of the following functions. State the domain of the function, identify 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?

Page 147

Ellen TurnellText BoxAdditional problems taken from Stitz and Zeager

Suggested assignment: p. 77: 3 (a-j)


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

Page 148




LturnellText Box Cube Root Function

LturnellText Box

LturnellStamp

LturnellPlaced Image

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

Page 162

-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


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

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 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?

Page 172



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

Page 173



(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

Page 174



(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

Page 175



(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

Page 176


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

Page 179

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

Page 200


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

Page 201

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

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 = 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

Page 216

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

Page 217

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)

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 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

Page 234



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 .


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.


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




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,)

Page 236


LturnellRectangle

LturnellRectangle

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

Page 254


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 )

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 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

Page 269

Quadratic Functions-Worksheet

Find the vertex and a and then use to sketch the graph of each function. Find the intercepts, axis of symmetry, and range of each function. Remember the domain is ( , ) 1. 2( ) 4 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 )



LturnellStamp

LturnellStamp

LturnellStamp

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

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 + +

Page 300




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

Page 313




LturnellText BoxFor an alternate method for 29-48 please see page 337



LturnellTypewritten Text




LturnellRectangle

LturnellStamp

LturnellRectangle

LturnellStamp

LturnellStamp

LturnellRectangle

LturnellStamp

LturnellRectangle

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

Page 326




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

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

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( ) 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

Page 338

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.

weltman revised (1)

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