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.
Alg II – Functions ~2~ NJCTL.org
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
Alg II – Functions ~3~ NJCTL.org
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)
Alg II – Functions ~4~ NJCTL.org
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.
Alg II – Functions ~5~ NJCTL.org
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
Alg II – Functions ~6~ NJCTL.org
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.
Alg II – Functions ~7~ NJCTL.org
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.
Alg II – Functions ~8~ NJCTL.org
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|
Alg II – Functions ~9~ NJCTL.org
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
Alg II – Functions ~10~ NJCTL.org
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)
Alg II – Functions ~11~ NJCTL.org
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)
Alg II – Functions ~12~ NJCTL.org
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)
Alg II – Functions ~13~ NJCTL.org
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)
Alg II – Functions ~14~ NJCTL.org
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).
Alg II – Functions ~15~ NJCTL.org
143. 𝑓(𝑥) = √1 − 𝑥23 ∗ 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).
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
Alg II – Functions ~16~ NJCTL.org
Inverse Functions Homework (2 pages) 149. 𝑓(𝑥) = 5𝑥 + 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).
150. 𝑓(𝑥) =2
3𝑥2 − 6 ∗ 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).
Alg II – Functions ~17~ NJCTL.org
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
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).
Alg II – Functions ~18~ NJCTL.org
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
Alg II – Functions ~19~ NJCTL.org
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)
Alg II – Functions ~20~ NJCTL.org
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).
Alg II – Functions ~21~ NJCTL.org
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, ∞)
Alg II – Functions ~22~ NJCTL.org
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
107. 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)
110. 𝐷: All Real Numbers
𝑅: 𝑦 ≤ 0
111. 𝐷: All Real Numbers
𝑅: All Real Numbers
112. 𝐷: All Real Numbers
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
𝑅: All Real Numbers
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.
Alg II – Functions ~23~ NJCTL.org
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)
Alg II – Functions ~24~ NJCTL.org
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: (−∞, ∞)
Alg II – Functions ~25~ NJCTL.org
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: (−∞, ∞)