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1. n(A) = 4 and n(B) = 3. Find n(A × B) and the number of relations from A to B.
12 and 2¹² = 4096.
2. A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : x − y is odd}. Write R in roster form.
{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}.
3. R = {(x, x³) : x is a prime less than 10}. Write R and its range.
R = {(2, 8), (3, 27), (5, 125), (7, 343)}; range {8, 27, 125, 343}.
4. R = {(1, 2), (2, 4), (3, 6)}. Find R⁻¹ and its domain and range.
R⁻¹ = {(2, 1), (4, 2), (6, 3)}; domain {2, 4, 6}, range {1, 2, 3}.
5. n(A) = 3 and n(B) = 2. The number of relations from A to B is: ((a) 6 (b) 8 (c) 32 (d) 64)
Option (d): 64.
6. How many reflexive relations are there on a set of 3 elements?
2^(9 − 3) = 64.
7. R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} on {1, 2, 3}. Which properties hold?
Reflexive only: not symmetric ((1, 2) without (2, 1)), not transitive ((1, 3) missing).
8. On the lines in a plane, L₁ R L₂ iff L₁ ⊥ L₂. Classify R.
Symmetric only: no line is perpendicular to itself, and L₁ ⊥ L₂, L₂ ⊥ L₃ give L₁ ∥ L₃.
9. How many symmetric relations are there on a set of 4 elements?
2^((16 + 4)/2) = 2¹⁰ = 1024.
10. A = {1, 2, 3}. The number of relations on A that are both reflexive and symmetric is: ((a) 8 (b) 56 (c) 64 (d) 512)
Option (a): 8.
11. Show R = {(a, b) : 3 divides a − b} on Z is an equivalence relation and give its classes.
Reflexive (0), symmetric, transitive (sums of multiples of 3); classes [0], [1], [2] by remainder.
12. How many equivalence relations are there on a 5-element set?
B₅ = 1 + 4 + 12 + 20 + 15 = 52.
13. A = {1, 2, 3}. Equivalence relations containing both (1, 2) and (2, 3)?
1: all three elements must be in one class.
14. R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} on {1, 2, 3}. Equivalence classes?
{1, 2} and {3}.
15. The number of equivalence relations on the set {1, 2, 3, 4} is: ((a) 5 (b) 14 (c) 15 (d) 16)
Option (c): 15.
16. Write {x : −3 ≤ x < 5} as an interval and give its length.
[−3, 5); length 8.
17. A = [−2, 4), B = (1, 6]. Find A ∩ B and A − B.
(1, 4) and [−2, 1].
18. From {1, 2, 3} to {a, b}: R₁ = {(1, a), (2, a)}, R₂ = {(1, a), (2, b), (3, a)}, R₃ = {(1, a), (1, b), (2, a), (3, b)}. Which are functions?
Only R₂.
19. Largest domain of f(x) = √(4 − x).
(−∞, 4].
20. A = {1, 2, 3}, B = {4, 5}. Which of these is a function from A to B? ((a) {(1, 4), (2, 5)} (b) {(1, 4), (1, 5), (2, 4), (3, 5)} (c) {(1, 5), (2, 5), (3, 5)} (d) {(1, 4), (2, 5), (3, 6)})
Option (c): {(1, 5), (2, 5), (3, 5)}.
21. f(x) = √(x − 1), g(x) = √(5 − x). Domains of f + g and f/g?
[1, 5] and [1, 5).
22. f(x) = 2x + 1, g(x) = x². Find fog and gof.
fog = 2x² + 1; gof = (2x + 1)².
23. f(x) = 1/(x − 1). Find fof and its domain.
(x − 1)/(2 − x); domain R − {1, 2}.
24. f(x) = √x, g(x) = 1 − x². Domain of fog?
[−1, 1].
25. f(x) = √x and g(x) = x² − 1. The domain of fog is: ((a) R (b) [1, ∞) (c) (−∞, −1] ∪ [1, ∞) (d) [−1, 1])
Option (c): (−∞, −1] ∪ [1, ∞).
26. Is f : R → R, f(x) = x³ one-one? Onto?
Both (strictly increasing; every real is a cube).
27. Show f : N → N, f(x) = 2x is one-one but not onto.
2x₁ = 2x₂ ⇒ x₁ = x₂; odd numbers have no pre-image.
28. Classify f : R → R, f(x) = x² + 2x + 3.
Many-one (f(−2) = f(0) = 3) and into (range [2, ∞)).
29. Find the codomain that makes f(x) = 2/(1 + x²), x ∈ R, onto.
(0, 2].
30. f : R → [0, ∞), f(x) = x². Then f is: ((a) one-one and onto (b) many-one and onto (c) one-one and into (d) many-one and into)
Option (b): many-one and onto.
31. One-one functions from {1, 2, 3} to {a, b, c, d, e}?
⁵P₃ = 60.
32. Onto functions from a 4-element set to a 3-element set?
81 − 48 + 3 = 36.
33. f : A → B, g : B → C, gof one-one. Must g be one-one?
No; only f must be.
34. Bijections from a 5-element set to itself?
5! = 120.
35. The number of onto functions from {1, 2, 3, 4} to {a, b} is: ((a) 16 (b) 14 (c) 12 (d) 2)
Option (b): 14.
36. Find the inverse of f(x) = (2x + 3)/(x − 1).
f⁻¹(x) = (x + 3)/(x − 2), x ≠ 2.
37. f : [0, ∞) → [−4, ∞), f(x) = x² − 4. Find f⁻¹.
√(x + 4).
38. f(x) = x³ + 1. Find f⁻¹(9).
2.
39. f(x) = 2x + 3, g(x) = x − 1. Find (gof)⁻¹(x).
(x − 2)/2.
40. f : [2, ∞) → [1, ∞), f(x) = x² − 4x + 5. Then f⁻¹(x) is: ((a) 2 + √(x − 1) (b) 2 − √(x − 1) (c) 2 ± √(x − 1) (d) 1/(x² − 4x + 5))
Option (a): 2 + √(x − 1).
41. If −2 ≤ x ≤ 3, find the ranges of x² and 1/x (x ≠ 0).
x² ∈ [0, 9]; 1/x ∈ (−∞, −1/2] ∪ [1/3, ∞).
42. Solve (x − 2)/(x + 3) ≥ 0.
x < −3 or x ≥ 2.
43. Solve (x + 1)²(x − 4)/(x − 1) < 0.
1 < x < 4.
44. Solve x² − 5x + 6 > 0.
x < 2 or x > 3.
45. The solution set of (x − 1)²(x + 2)/(x − 3) ≤ 0 is: ((a) [−2, 3) (b) [−2, 3] (c) [−2, 1) ∪ (1, 3) (d) (−∞, −2] ∪ (3, ∞))
Option (a): [−2, 3).
46. Range of x² − 6x + 10.
[1, ∞).
47. Domain and range of √(x² − 6x + 8).
(−∞, 2] ∪ [4, ∞); [0, ∞).
48. Range of (x² + 1)/(x² + 2).
[1/2, 1).
49. Range of x/(x² + 1).
[−1/2, 1/2].
50. For which k is x² + kx + 4 > 0 for every real x?
−4 < k < 4.
51. The range of f(x) = (x² + x + 2)/(x² + x + 1), x ∈ R, is: ((a) (1, 7/3] (b) [1, 7/3] (c) (1, ∞) (d) [7/3, ∞))
Option (a): (1, 7/3].
52. Solve |x − 3| = 5.
x = 8 or x = −2.
53. Solve |2x − 1| < 3.
−1 < x < 2.
54. Range of |x − 1| for −2 ≤ x < 4.
[0, 3].
55. Range of f(x) = |x| − x.
[0, ∞).
56. The domain of f(x) = 1/√(|x| − x) is: ((a) (−∞, 0) (b) (0, ∞) (c) (−∞, 0] (d) R)
Option (a): (−∞, 0).
57. Range of 3 sin x + 4 cos x − 2.
[−7, 3].
58. Domain of sec x + cosec x.
R − {nπ/2}.
59. General solution of cos x = 1/2.
x = 2nπ ± π/3.
60. On [0, 2π], where are sin x and cos x both ≥ 0?
[0, π/2].
61. The range of 5 sin x − 12 cos x + 7 is: ((a) [−6, 20] (b) [−13, 13] (c) [−10, 24] (d) [2, 12])
Option (a): [−6, 20].
62. Domain of √(sin x).
∪[2nπ, (2n + 1)π].
63. Range of cos²x + cos x + 1.
[3/4, 3].
64. Range of 1/(3 + sin x).
[1/4, 1/2].
65. Domain of log(cos x).
∪(2nπ − π/2, 2nπ + π/2).
66. The number of real solutions of sin x = x/5 is: ((a) 1 (b) 3 (c) 5 (d) 7)
Option (b): 3.
67. Find cos⁻¹(cos 7π/6).
5π/6.
68. Find tan⁻¹(−√3) + cot⁻¹(−1/√3).
−π/3 + 2π/3 = π/3.
69. Domain of sin⁻¹(2x − 1).
[0, 1].
70. If sin⁻¹x = π/5, find cos⁻¹x.
3π/10.
71. The value of sin⁻¹(sin(2π/3)) is: ((a) 2π/3 (b) π/3 (c) −π/3 (d) π/6)
Option (b): π/3.
72. Domain of log₂(x² − 4).
(−∞, −2) ∪ (2, ∞).
73. Solve log base 1/3 of x > 2.
0 < x < 1/9.
74. Range of 3^(x²).
[1, ∞).
75. Domain of log base x of (5 − x).
(0, 1) ∪ (1, 5).
76. The solution set of log₀.₅(x − 2) > log₀.₅ 4 is: ((a) (2, 6) (b) (6, ∞) (c) (−∞, 6) (d) (2, ∞))
Option (a): (2, 6).
77. Find [−3.7], {−3.7} and [π] + [−π].
−4; 0.3; −1.
78. Solve [x] = 2.5.
No solution: [x] is always an integer.
79. Solve 2[x] = x + {x}.
x = 0 or x = 3/2.
80. Domain of 1/√([x] − 1).
[2, ∞).
81. Minimum value of max{x, 1 − x}.
1/2, at x = 1/2.
82. The solution set of [x]² + [x] − 2 = 0 ([·] = greatest integer function) is: ((a) {−2, 1} (b) [−2, −1) ∪ [1, 2) (c) [−2, 2) (d) (−2, −1] ∪ (1, 2])
Option (b): [−2, −1) ∪ [1, 2).
83. Is log(x + √(x² + 1)) even or odd?
Odd.
84. Even and odd parts of x² + 3x + 2.
x² + 2 and 3x.
85. Is (2ˣ + 1)/(2ˣ − 1) even or odd?
Odd.
86. f(x) = x(eˣ − 1)/(eˣ + 1) is: ((a) odd (b) even (c) neither even nor odd (d) both even and odd)
Option (b): even.
87. Period of sin(4x/3).
3π/2.
88. Period of sin 3x + cos 2x.
2π.
89. Period of tan x + cot x.
π.
90. Period of {3x} + sin(πx).
2.
91. The fundamental period of |sin 2x| + |cos 2x| is: ((a) π/4 (b) π/2 (c) π (d) 2π)
Option (a): π/4.
92. f(x + y) = f(x) + f(y), f(2) = 10. Find f(7).
35.
93. f(xy) = f(x) + f(y), f(3) = 2. Find f(81) and f(1/9).
8 and −4.
94. 2f(x) + f(1 − x) = x². Find f(x).
f(x) = (x² + 2x − 1)/3.
95. f(x + y) = f(x) f(y) for all real x, y, and f(1) = 3. Then f(1) + f(2) + … + f(n) equals: ((a) 3(3ⁿ − 1)/2 (b) 3ⁿ − 1 (c) 3n(n + 1)/2 (d) (3ⁿ − 1)/2)
Option (a): 3(3ⁿ − 1)/2.
96. How is y = 3 − 2f(x) obtained from y = f(x)?
Stretch vertically by 2, reflect in the x-axis, then shift up 3.
97. Sketch y = |x² − 4| and give its range.
Parabola with the part between −2 and 2 flipped up; range [0, ∞).
98. How is y = f(2x − 4) obtained from y = f(x)?
f(2(x − 2)): squeeze by 2 towards the y-axis, then shift right 2.
99. Number of solutions of |x² − 4| = 3.
4 (x² = 7 or x² = 1).
100. The graph of y = f(x + 3) − 2 is obtained from y = f(x) by shifting it: ((a) 3 left, 2 down (b) 3 right, 2 down (c) 3 left, 2 up (d) 3 right, 2 up)
Option (a): 3 left, 2 down.