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Alg II – Functions ~1~ NJCTL.org

Relations and Functions – Class Work Name _________________________________ Is the relation a function? 1. {(1,2), (3,4), (5,6)} 2. {(4,3), (3,2), (4,2)} 3. {(5,1), (3,1), (−4,1)}

4. 5. 6.

7. 8. 9.

10.

11.

12. 13.


Relations and Functions – Home Work Is the relation a function? 14. {(3,1), (−2,6), (1,4)} 15. {(1,2), (2,2), (1,2)} 16. {(2,1), (5,1), (−6,7)}

17. 18. 19.

20. 21. 22.

23.

24.

25. 26.

Spiral Review Simplify each of the following

27. (𝑥4)−3 ∙ 2𝑥4 28. 2𝑥2𝑦4∙4𝑥2𝑦4∙3𝑥

3𝑥−3𝑦2 29. (2𝑥3𝑧2)

3

𝑥3𝑦4𝑧2∙𝑥−4𝑧3


Evaluating Functions – Class Work Name _________________________________ Find the following: 30. If 𝑓(𝑥) = 3𝑥 + 4,

𝐹𝑖𝑛𝑑 𝑓(2)

31. If 𝑓(𝑥) = −√𝑥 − 3,𝐹𝑖𝑛𝑑 𝑓(19) If

32. If ℎ(𝑥) = |𝑥 − 4|,𝐹𝑖𝑛𝑑 ℎ(−6)

33. If 𝑔(𝑥) = 3𝑥3,𝐹𝑖𝑛𝑑 𝑔(−2)

34. If 𝑓(𝑥) = 2𝑥2 − 2,𝐹𝑖𝑛𝑑 𝑓(2 − 𝑎)

35. If ℎ(𝑥) = (𝑥 − 2)2 + 2,𝐹𝑖𝑛𝑑 ℎ(2𝑏 + 1)

36. If 𝑔(𝑥) = 2𝑥2 − 𝑥,𝐹𝑖𝑛𝑑 𝑔(𝑚 − 2)

37. If 𝑓(𝑥) =1

2𝑥 + 3,

𝐹𝑖𝑛𝑑 𝑓(4𝑥2)

Evaluating Functions – Home Work Find the following:

38. If 𝑓(𝑥) = (𝑥 − 1)2,𝐹𝑖𝑛𝑑 𝑓(−5)

39. If 𝑓(𝑥) = −|2𝑥 − 3|,𝐹𝑖𝑛𝑑 𝑓(−4) If

40. If ℎ(𝑥) = 𝑥3 − 1,𝐹𝑖𝑛𝑑 ℎ(−2)

41. If 𝑔(𝑥) = −2𝑥2 − 1,𝐹𝑖𝑛𝑑 𝑔(4)

42. If 𝑓(𝑥) = −3𝑥 + 2,𝐹𝑖𝑛𝑑 𝑓(−𝑥 − 6)

43. If ℎ(𝑥) = (2𝑥 − 1)2,𝐹𝑖𝑛𝑑 ℎ(1 − 2𝑝)

44. If 𝑔(𝑥) = 𝑥3 − 𝑥,𝐹𝑖𝑛𝑑 𝑔(𝑎2)

45. If 𝑓(𝑥) =8

𝑥2 ,

𝐹𝑖𝑛𝑑 𝑓(2𝑚)

Spiral Review

Multiply each of the following 46. (4𝑥 + 1)(2𝑥 + 6) 47. (7𝑥 − 6)(5𝑥 + 6) 48. (𝑥2 + 6𝑥 − 4)(2𝑥 − 4)


Interval and Inequality Notation – Class Work Name _________________________________ Give the interval and inequality notation for each graph. 49.

50. 51. 52. 53. 54. 55. 56. 57. 58.


Interval and Inequality Notation – Homework Give the interval and inequality notation for each graph. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.

Spiral Review Simplify each of the following

69. (𝑥−2 ∙ 𝑥−3)4 70. 6𝑥2𝑦2

3𝑥−1∙4𝑦𝑥2 71. (2𝑝𝑚−1𝑞0)

−42𝑚−1𝑝3

2𝑝𝑞2


Discrete vs. Continuous – Class Work Name _________________________________ Is the relation discrete or continuous? If continuous, state the interval of continuity. 72. {(1,2), (3,4), (5,6)} 73. {(4,3), (3,2), (4,2)} 74. {(5,1), (3,1), (−4,1)}

75. 76. 77.

78. 79. 80.

81. 82.

83. 84.


Discrete vs. Continuous – Home Work Is the relation discrete or continuous? If continuous, state the interval of continuity. 85. {(3,1), (−2,6), (1,4)} 86. {(1,2), (2,2), (1,2)} 87. {(2,1), (5,1), (−6,7)}

88. 89. 90.

91. 92. 93.

94. 95.

96. 97.


Domain and Range – Class Work Name _________________________________ Find the domain and range for each of the following. Write your answers in interval notation where

appropriate. 98. {(1,2), (3,4), (5,6)} 99. {(4,3), (3,2), (4,2)} 100.

101. 102. 103.

104. 105. 106.

For some functions below, you may need to use a graphing calculator or on an online program (like desmos.com) to find the range.

107. 𝑓(𝑥) = −2

3𝑥 − 3 108. 𝑔(𝑥) = √2 − 𝑥 109. ℎ(𝑥) =

1

√2𝑥−5

110. 𝑔(𝑥) = −|𝑥 − 2| 111. 𝑓(𝑥) = (𝑥 − 2)3 + 1 112. ℎ(𝑥) = |𝑥2|


Domain and Range – Home Work Find the domain and range for each of the following. Write your answers in interval notation where

appropriate. 113. {(3,1), (−2,6), (1,4)} 114. {(1,2), (2,2), (1,2)} 115.

116. 117. 118.

119. 120. 121.

For some functions below, you may need to use a graphing calculator or on an online program (like desmos.com) to find the range.

122. ℎ(𝑥) = −2

3𝑥2

123. 𝑔(𝑥) = −√2𝑥 − 1 124. 𝑓(𝑥) =1

√3𝑥−2

125. ℎ(𝑥) = 3𝑥2 − 𝑥 + 2 126. 𝑓(𝑥) =2𝑥−3

5 127. 𝑔(𝑥) =

−2𝑥

√3𝑥+4


Operations with Functions Name _________________________________ Class Work Given: and

109. Given that 𝑓(𝑥) = 3𝑥2 − 4, 𝑔(𝑥) = |3𝑥 − 2| − 1, and ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥).

Find: a) h(x) b) h(2) c) h(0) d) the domain of h(x) 110. Given that 𝑓(𝑥) = (2𝑥 − 3), 𝑔(𝑥) = −3𝑥2, and ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥).

Find: a) h(x) b) h(-2) c) h(1) d) the domain of h(x)

111. Given that 𝑓(𝑥) = √𝑥 − 3, 𝑔(𝑥) = −2𝑥2, and ℎ(𝑥) =𝑓(𝑥)

𝑔(𝑥).

Find: a) h(x) b) h(2a) c) h(m - 2) d) the domain of h(x) 112. Given that 𝑓(𝑥) = 2 − 𝑥, 𝑔(𝑥) = 3𝑥 − 2, and ℎ(𝑥) = 2𝑓(𝑥) − 3𝑔(𝑥).

Find: a) h(x) b) h(-4p) c) h(1 - k) d) the domain of h(x)

113. Given that 𝑓(𝑥) = 3𝑥 + 1, 𝑔(𝑥) = √𝑥 − 2 ℎ(𝑥) = 𝑓(𝑥)

(𝑔(𝑥))2

Find: a) h(x) b) h(2a) c) h(1 - p) d) the domain of h(x)


Operations with Functions Homework

118. Given 𝑓(𝑥) = √𝑥 + 5, 𝑔(𝑥) = (2𝑥 + 1)2 and ℎ(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)

Find: a) h(x) b) h(4) c) h(-5) d) the domain of h(x) 119. Given 𝑓(𝑥) = 2 − 𝑥, 𝑔(𝑥) = 4 − 𝑥 and ℎ(𝑥) = 𝑔(𝑥)𝑓(𝑥)

Find: a) h(x) b) h(2) c) h(0) d) the domain of h(x)

120. Given 𝑓(𝑥) = √𝑥 − 5, 𝑔(𝑥) = |𝑥 + 2| and ℎ(𝑥) =𝑔(𝑥)

𝑓(𝑥)

Find: a) h(x) b) h(30) c) h(3k - 2) d) the domain of h(x) 121. Given 𝑓(𝑥) = −2𝑥3 − 1, 𝑔(𝑥) = 2𝑥2 − 𝑥 and ℎ(𝑥) = 5𝑓(𝑥) − 2𝑔(𝑥)

Find: a) h(x) b) h(a2) c) h(- m) d) the domain of h(x)

122. Given 𝑓(𝑥) = 2𝑥 − 3, 𝑔(𝑥) = √2 − 3𝑥, and ℎ(𝑥) = −𝑓(𝑥)

(𝑔(𝑥))2

Find: a) h(x) b) h(1 - x) c) h(2b) d) the domain of h(x)

Spiral Review 123. Graph: 124. Graph: 125. Factor: 126. Multiply:

𝑦 = −|𝑥 − 3| + 5 𝑦 = √−𝑥 + 2 − 4 16x2 – 81 (2x + 3)(4x2 + 2)


Composite Functions Name _________________________________ Class Work 127. 𝑓(𝑥) = 3𝑥 − 2; 𝑔(𝑥) = −2𝑥 + 4

Find: a) 𝑔 ∘ 𝑓 b) (𝑔 ∘ 𝑓)(−3) 128. 𝑓(𝑥) = 𝑥2 + 1; 𝑔(𝑥) = 5𝑥 − 1

Find: a) f(g(x)) b) f(g(0)).

129. 𝑓(𝑥) =2

𝑥−2; 𝑔(𝑥) = 2𝑥2 − 9

Find: a) 𝑓 ∘ 𝑔 b) (𝑓 ∘ 𝑔)(6)

130. 𝑓(𝑥) =𝑥

𝑥2−1+ 3; 𝑔(𝑥) = √𝑥 + 2

Find: a) f(g(x)) b) f(g(-2)). 131. 𝑓(𝑥) = 𝑥3; 𝑔(𝑥) = |𝑥 − 1|

Find: a) 𝑔 ∘ 𝑓 b) (𝑔 ∘ 𝑓)(−2)


Composite Functions Homework

132. 𝑓(𝑥) = −1

2𝑥 + 3; 𝑔(𝑥) = −4𝑥 + 2

Find: a) 𝑔 ∘ 𝑓 b) (𝑔 ∘ 𝑓)(2) 133. 𝑓(𝑥) = 2𝑥2 − 5; 𝑔(𝑥) = 2𝑥 + 3

Find: a) f(g(x)) b) f(g(-1))

134. 𝑓(𝑥) =−1

𝑥+2; 𝑔(𝑥) = 3𝑥2 − 10

Find: a) f(g(x)) b) f(g(0))

135. 𝑓(𝑥) =𝑥

𝑥2+4+ 3; 𝑔(𝑥) = √2 − 𝑥

Find: a) f(g(x)) b) f(g(-3)) 136. 𝑓(𝑥) = 𝑥2 − 4; 𝑔(𝑥) = |𝑥 − 1|

Find: a) 𝑔 ∘ 𝑓 b) (𝑔 ∘ 𝑓)(3) Spiral Review 137. Graph: 138. Graph: 139. Simplify: 140. Multiply:

𝑦 = −log (𝑥 + 3) 𝑦 = 2(𝑥 + 2)2 − 4 −5𝑥3𝑦−3

25𝑥−7𝑦−2 (-2m2n3)(-8m5n4)


Inverse Functions Name _________________________________ Class Work (2 pages) 141. 𝑓(𝑥) = 3𝑥 − 2

i) Given f(x), find f-1(x) ii) Algebraically show that f(f-1(x)) = f-1(f(x)) = x. iii) Graph f(x) and f-1(x) on the same graph. iv) Describe the domain and range for f -1(x). 142. 𝑓(𝑥) = 2𝑥2 + 1 ∗ 𝑑𝑜𝑚𝑎𝑖𝑛 𝑖𝑠 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑 𝑡𝑜 [0, ∞)

i) Given f(x), find f-1(x) ii) Algebraically show that f(f-1(x)) = f-1(f(x)) = x. iii) Graph f(x) and f-1(x) on the same graph. iv) Describe the domain and range for f -1(x).


143. 𝑓(𝑥) = √1 − 𝑥23 ∗ 𝑑𝑜𝑚𝑎𝑖𝑛 𝑖𝑠 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑 𝑡𝑜 [0, ∞)


144. 𝑓(𝑥) =3

4𝑥−2

i) Given f(x), find f-1(x) ii) Algebraically show that f(f-1(x)) = f-1(f(x)) = x. iii) Graph f(x) and f-1(x) on the same graph. iv) Describe the domain and range for f -1(x). Spiral Review 145. Find: f ◦ g 146. Factor: 147. Simplify 148. Graph: If g(x) = x2 + 2 16x2 –25y2 (-2x3y2)4 𝑦 = −|3𝑥| + 2 and f(x) = (x – 1)2


Inverse Functions Homework (2 pages) 149. 𝑓(𝑥) = 5𝑥 + 2


150. 𝑓(𝑥) =2

3𝑥2 − 6 ∗ 𝑑𝑜𝑚𝑎𝑖𝑛 𝑖𝑠 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑 𝑡𝑜 [0, ∞)



151. 𝑓(𝑥) = √𝑥 − 4

i) Given f(x), find f-1(x) ii) Algebraically show that f(f-1(x)) = f-1(f(x)) = x. iii) Graph f(x) and f-1(x) on the same graph. iv) Describe the domain and range for f -1(x). 152. 𝑓(𝑥) = −3𝑥 + 2



Unit Review Multiple Choice – Determine the best answer for each question.

1. Determine the domain of {(1,3), (5,6), (6,8)}

a. {1, 5, 8} b. {1, 5, 6} c. {3, 6, 8} d. Set of Reals

2. Determine the range of 𝑓(𝑥) = |𝑥 − 2| + 3.

a. [3, ∞] b. [1, ∞) c. (1, ∞) d. (3, ∞)

3. What is the domain of the graph to the right?

a. −10 ≤ 𝑥 ≤ 10

b. −10 < 𝑥 < 10

c. −6 ≤ 𝑥 ≤ −2 or 0 ≤ 𝑥 ≤ −6

d. −10 ≤ 𝑥 ≤ −4 or −2 ≤ 𝑥 ≤ 4 or 6 ≤ 𝑥 ≤ 10

4. Which choice represents a discrete set?

a. The time it takes people to tie their shoes. b. The amount of rain in a given week.

c. The number of people attending a play. d. The number of rotations of a wheel.

5. Which of the following is a function?

a. 𝑥2 + 𝑦2 = 4 b. 𝑥 + 𝑦2 = 4 c. 𝑥2 + 𝑦 = 4 d. 4𝑥2 + 𝑦2 = 4

6. 𝑓(𝑥) = 3𝑥2 − 2 , 𝑔(𝑥) = 4 − 𝑥, and ℎ(𝑥) = 𝑓(𝑥) − 𝑔(𝑥). h(3) =

a. 78

b. 26

c. 24

d. 18

7. 𝑓(𝑥) = 3𝑥2 − 2 , 𝑔(𝑥) = 4 − 2𝑥, and ℎ(𝑥) = 𝑓(𝑥)/𝑔(𝑥). h(3) =

a. -50

b. -25

c. 25

d. -12.5

8. 𝑓(𝑥) = (3𝑥)2 − 4 , 𝑔(𝑥) = 5 − 4𝑥, and ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥). h(3) =

a. -539

b. -161

c. -7

d. 7

9. 𝑓(𝑥) = 3𝑥2 − 2 , 𝑔(𝑥) = 4 − 𝑥, and ℎ(𝑥) = 𝑓(𝑔(𝑥)). h(3) =

a. -5

b. -3

c. -1

d. 1


10. 𝑓(𝑥) = 3𝑥2 − 2 , 𝑔(𝑥) = 4 − 𝑥, and ℎ(𝑥) = 𝑔(𝑓(𝑥)). h(2a + 1) =

a. −12𝑎2 − 12𝑎 + 3

b. 25 − 36𝑎 + 12𝑎2

c. −6𝑎2 + 5

d. 12𝑎2 + 3

11. Given 𝑓(𝑥) = 2𝑥3 − 2, find 𝑓−1(8)

a. -1022

b. √53

c. 1

2

d. 2

12. Given 𝑓(𝑥) = 2𝑥3 − 2 and 𝑓−1(𝑎) = −3, find a.

a. -27

b. −√33

c. -56

d. 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

Short Constructed Response

13) Evaluate the function at all of the given points. 𝑦 = 2√𝑥 − 9 + 3 a) f(25) b) f(9) c) f(10) d) f(3x – 4)

14) Find 𝑓 + 𝑔, 𝑓 – 𝑔, 𝑓𝑔 𝑎𝑛𝑑 𝑓/𝑔 for the following functions. Then, find their domain. 𝑓(𝑥) = 𝑥 − 1, g(x) = 3x2 + 2 15) Evaluate the function for f(x) = x2 – 1 and g(x) = x + 2 a) (f + g)(2) b) (f – g)(-1) c) (fg)(3) d) (f/g)(0)


16) Find f ◦ g and g ◦ f: a) f(x) = x2, g(x) = x + 3 b) f(x) = 2x – 5, g(x) = x2 + 2 17) Determine whether the function has an inverse function. If it does, find the inverse function.

a) f(x) = 3x2 *domain restricted to [0, ∞) b) 𝑓(𝑥) = √𝑥 + 13

c) f(x) = 1/x d) f(x) = 2x + 3 Extended Response 18. 𝑓(𝑥) = (𝑥 − 2)3 + 4

i) Given f(x), find f-1(x) ii) Algebraically show that f(f-1(x)) = f-1(f(x)) = x. iii) Using technology to help you, graph f(x) and f-1(x) on the same graph. iv) Describe the domain and range for f -1(x).


Relations and Functions Classwork

1. Function

2. Not a Function

3. Function

4. Function

5. Function

6. Not a Function

7. Not a Function

8. Not a Function

9. Function

10. Function

11. Function

12. Function

13. Not a Function

Relations and Function Homework

14. Function

15. Function

16. Function

17. Function

18. Not a Function

19. Not a Function

20. Function

21. Not a Function

22. Not a Function

23. Function

24. Not a Function

25. Function

26. Function 27. 28. 29. Evaluating Functions Class Work 30. 𝑓(2) = 10

31. 𝑓(19) = −4

32. ℎ(−6) = 10

33. 𝑔(−2) = −24

34. 𝑓(2 − 𝑎) = 2𝑎2 − 8𝑎 + 6

35. ℎ(2𝑏 + 1) = 4𝑏2 − 4𝑏 + 3

36. 𝑔(𝑚 − 2) = 2𝑚2 − 9𝑚 +10

37. 𝑓(4𝑥2) =1

8𝑥2+3

Evaluating Functions Class Work

38. 𝑓(−5) = 36

39. 𝑓(−4) = −11

40. ℎ(−2) = −9 41. 𝑔(4) = −33

42. 𝑓(−𝑥 − 6) = 3𝑥 + 20

43. ℎ(1 − 2𝑝) = 16𝑝2 − 8𝑝 +1

44. 𝑔(𝑎2) = 𝑎6 − 𝑎2

45. 𝑓(2𝑚) =2

𝑚2

46. 8𝑥2 + 26𝑥 + 6

47. 35𝑥2 + 12𝑥 − 36

48. 2𝑥3 + 8𝑥2 − 32𝑥 + 16 Interval and Inequality Notation Classwork

49. [1, ∞)

𝑥 ≥ 1

50. (−∞, −3) 𝑥 < −3

51. [−2, 6] −2 ≤ 𝑥 ≤ 6

52. [−3, 1) −3 ≤ 𝑥 < 1

53. (1, 9) 1 < 𝑥 < 9

54. (−∞, 0] 𝑥 ≤ 0

55. [0, ∞) 𝑥 ≥ 0

56. [−8, −4] or [2, ∞) −8 ≤ 𝑥 ≤ −4 or 𝑥 ≥ 2

57. (−∞, −7] 𝑜𝑟 (−5, ∞) 𝑥 ≤ −7 𝑜𝑟 𝑥 > −5

58. (−∞, −5] 𝑜𝑟 (4, ∞) 𝑥 ≤ −5 𝑜𝑟 𝑥 > 4

Interval and Inequality Notation Homework

59. [−4, ∞) 𝑥 ≥ −4

60. (−∞, 2) 𝑥 < 2

61. [−5, 3] −5 ≤ 𝑥 ≤ 3

62. [2, 6) 2 ≤ 𝑥 < 6

63. (−8, 0) −8 < 𝑥 < 0

64. (−∞, 5] 𝑥 ≤ 5

65. [−9, ∞) 𝑥 ≥ −9

66. [−4, 0] or (5, ∞) −4 ≤ 𝑥 ≤ 0 or 𝑥 > 5

67. (−∞, −4) 𝑜𝑟 (2, ∞) 𝑥 < −4 𝑜𝑟 𝑥 > 2

68. (−6, 0] or (3, ∞) −6 < 𝑥 ≤ 0 𝑜𝑟 𝑥 > 3

Spiral Review

69. 1

𝑥20

70. 2𝑥𝑦

71. 16𝑥6

𝑦5𝑧2

Discrete vs. Continuous Classwork

72. Discrete

73. Discrete

74. Discrete

75. Discrete

76. Discrete

77. Discrete

78. Discrete

79. Discrete

80. Discrete

81. Discrete

82. Continuous [−4, ∞)


83. Continuous (−∞, −2] or [2, ∞)

84. Continuous All Real Numbers

Discrete vs. Continuous Homework

85. Discrete

86. Discrete

87. Discrete

88. Discrete

89. Discrete

90. Discrete

91. Discrete

92. Discrete

93. Discrete

94. Discrete

95. Continuous [−6, 6]

96. Continuous [−8, 0)

97. Continuous All Real Numbers

Domain And Range Classwork

98. 𝐷: {1, 3, 5} 𝑅: {2, 4, 6}

99. 𝐷: {3,4} 𝑅: {2, 3}

100. 𝐷: {−4, 0, 2} 𝑅: {3, 4, 5}

101. 𝐷: {4, 5, 6} 𝑅: {6}

102. 𝐷: {−2, −1, 2, 3} 𝑅: {0, 3, 4, 5, 7}

103. 𝐷: {1, 2} 𝑅: {3, 4, 5, 6}

104. 𝐷: 𝑥 ≤ −2 or 𝑥 > 2 𝑅: All Real Numbers

105. 𝐷: All Real Numbers 𝑅: 𝑦 ≥ 2

106. 𝐷: All Real Numbers

𝑅: All Real Numbers



108. 𝐷: 𝑥 ≤ 2

𝑅: 𝑦 ≥ 0

109. 𝐷: 𝑥 >5

2

𝑅: 2.24 > 𝑦

> 0 (𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦)


𝑅: 𝑦 ≤ 0




𝑅: 𝑦 ≥ 0

Domain And Range

Classwork

113. 𝐷: {−2, 1, 3}

𝑅: {1, 4, 6}

114. 𝐷: {1, 2}

𝑅: {2}

115. 𝐷: {−1, 0, 1}

𝑅: {6, 7, 8}

116. 𝐷: {2, 4}

𝑅: {6, 7, 8}

117. 𝐷: {3, 4, 5, 6}

𝑅: {1, 2, 3, 4}

118. 𝐷: {5}

𝑅: {0, 1, 2, 3}

119. 𝐷: {−4, −2, 2, 4, 5}

𝑅: {−3, 2, 4, 5}

120. 𝐷: − 6 ≤ 𝑥 ≤ 6

𝑅: − 6 ≤ 𝑦 ≤ 6

121. 𝐷: 𝑥 ≥ −2


122. 𝐷: x ≠ 0

𝑅: 𝑦 < 0

123.

124. 125. (4x-9)(4x+9)

126. 8𝑥3 + 12𝑥2 + 4𝑥 + 6

127. a) −6𝑥 + 8

b) 26

128. a) 25𝑥2 − 10𝑥 + 2

b) 2

129. a) 2

2𝑥2−11

b) 2

61

130. a) √𝑥+2

𝑥+1

b) 0

131. a) |𝑥3 − 1|

b) 9

132. a) 2x - 10

b) 10

133. a) 8𝑥2 + 24𝑥 + 13

b) -3

134. a) −1

3𝑥2−8

b) 1

8

135. a) √2−𝑥

6−𝑥+ 3

b) √5

9+ 3

136. a) |𝑥2 − 5|

b) 4

137.


138.

139. −𝑥10

5𝑦

140. 16𝑚7𝑛7

141. 𝑖) 𝑓−1(𝑥) =𝑥+2

3

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = 3 (𝑥 + 2

3) − 2

= 𝑥 + 2 − 2 = 𝑥

𝑓−1(𝑓(𝑥)) =(3𝑥 − 2) + 2

3=

3𝑥

3= 𝑥

iii)

iv) The domain and range are both the set of all

real numbers.

142. 𝑖) 𝑓−1(𝑥) = √𝑥−1

2

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = 2 (√𝑥 − 1

2)

2

+ 1 = 2 (𝑥 − 1

2) + 1

= 𝑥 − 1 + 1 = 𝑥

𝑓−1(𝑓(𝑥)) = √(2𝑥2 + 1) − 1

2= √

2𝑥2

2= 𝑥

iii) iv) 𝐷: 𝑥 ≥ 1, 𝑅: 𝑦 ≥ 0

143. 𝑖) 𝑓−1(𝑥) = √1 − 𝑥3

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = √1 − (√1 − 𝑥3)23

=

√1 − (1 − 𝑥3)3

= √𝑥33= 𝑥

𝑓−1(𝑓(𝑥)) = √1 − ( √1 − 𝑥23)

3

=

√1 − (1 − 𝑥2) = √𝑥2 = 𝑥

iii) , dashed line is f-1, dotted line is f. iv) 𝐷: 𝑥 ≤ 1, 𝑅: 𝑦 ≥ 0

144. 𝑖) 𝑓−1(𝑥) =3

4𝑥+

1

2

𝑖𝑖) 𝑓(𝑓−1(𝑥)) =3

4 (3

4𝑥+

12

) − 2=

3

3𝑥

+ 2 − 2=

3

3𝑥

= 𝑥

𝑓−1(𝑓(𝑥)) =3

4 (3

4𝑥 − 2)

+1

2=

3

124𝑥 − 2

+1

2=

3(4𝑥 − 2)

12+

1

2=

12𝑥 − 6 + 6

12= 𝑥

iii) , dashed line

is f.

iv) 𝐷: 𝑥 ≠ 0, 𝑅: 𝑦 ≠1

2

145. 𝑓 ∘ 𝑔 = 𝑥4 + 2𝑥2 + 1

146. (4𝑥 − 5𝑦)(4𝑥 + 5𝑦)


147. 16𝑥12𝑦8

148.

149. 𝑖) 𝑓−1 =𝑥−2

5

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = 5 (𝑥 − 2

5) + 2 =

𝑥 − 2 + 2 = 𝑥

𝑓−1(𝑓(𝑥)) =5𝑥 + 2 − 2

5= 𝑥

iii)

iv) 𝐷: (−∞, ∞), 𝑅: (−∞, ∞)

150. 𝑖) 𝑓−1(𝑥) = √3

2𝑥 + 9

𝑖𝑖) 𝑓(𝑓−1(𝑥)) =2

3(√

3

2𝑥 + 9)

2

− 6 =

2

3(

3

2𝑥 + 9) − 6 =

𝑥 + 6 − 6 = 𝑥

𝑓−1(𝑓(𝑥)) = √3

2(

2

3𝑥2 − 6) + 9 =

√𝑥2 − 9 + 9 = √𝑥2 = 𝑥

iii)

iv) 𝐷: 𝑥 ≥ 6, 𝑅: 𝑦 ≥ 0

151. 𝑖) 𝑓−1(𝑥) = 𝑥2 + 4

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = √𝑥2 + 4 − 4 = √𝑥2 = 𝑥

𝑓−1(𝑓(𝑥)) = (√𝑥 − 4)2

+ 4 =

𝑥 − 4 + 4 = 𝑥

iii)

iv) 𝐷: 𝑥 ≥ 0, 𝑅: 𝑦 ≥ 4

152. 𝑖) 𝑓−1(𝑥) =−𝑥+2

3

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = −3 (−𝑥 + 2

3) + 2 =

𝑥 − 2 + 2 = 𝑥

𝑓−1(𝑓(𝑥)) =−(−3𝑥 + 2) + 2

3=

3𝑥 − 2 + 2

3=

3𝑥

3= 𝑥

iii)

iv) 𝐷: (−∞, ∞), 𝑅: (−∞, ∞)


Unit Review:

1. b

2. a

3. d

4. c

5. c

6. c

7. d

8. a

9. 9

10. a

11. b

12. c

13. a) 11, b) 3, c) 5, d) 2√3𝑥 − 4 + 3

14. 𝑓 + 𝑔 = 3𝑥2 + 𝑥 + 1, 𝐷: (−∞, ∞)

𝑓 − 𝑔 = −3𝑥2 + 𝑥 − 3, 𝐷: (−∞, ∞)

𝑓𝑔 = 3𝑥3 − 3𝑥2 + 2𝑥 − 1, 𝐷: (−∞, ∞)

𝑓

𝑔=

𝑥 − 1

3𝑥2 + 2, 𝐷: (−∞, ∞)

15. a) 7, b) -1, c) 40, d) -1/2

16. 𝑎) 𝑓 ∘ 𝑔 = 𝑥2 + 6𝑥 + 9, 𝑔 ∘ 𝑓 = 𝑥2 + 3

𝑏) 𝑓 ∘ 𝑔 = 2𝑥2 − 1, 𝑔 ∘ 𝑓 = 4𝑥2 − 2𝑥 + 27

17. a) yes, 𝑓−1(𝑥) = √𝑥

3

b) yes, 𝑓−1(𝑥) = 𝑥3 − 1

c) yes, 𝑓−1(𝑥) =1

𝑥

d) yes, 𝑓−1(𝑥) =𝑥−3

2

18. 𝑖) 𝑓−1(𝑥) = √𝑥 − 43

+ 2

𝑖𝑖) 𝑓(𝑓−1(𝑥)) = ( √𝑥 − 43

+ 2 − 2)3

+ 4 =

√𝑥 − 43 3

+ 4 = 𝑥 − 4 + 4 = 𝑥

𝑓−1(𝑓(𝑥)) = √(𝑥 − 2)3 + 4 − 43

+ 2 =

√(𝑥 − 2)33+ 2 = 𝑥 − 2 + 2 = 𝑥

iii)

iv) 𝐷: (−∞, ∞), 𝑅: (−∞, ∞)

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