1
For the function f(x) = √(x − (-16)), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − (-16) ≥ 0, so x ≥ -16. The smallest such value is -16.
2
An employee's salary starts at Rs. 75967 and increases by Rs. 3485 each year. What will the salary be after 15 years?
Medium
ExplanationSalary after 15 years = 75967 + 15 × 3485 = Rs. 128242.
3
If f(x) = 6x − 8 and g(x) = f(x − 5) + 2, what is g(9)?
Medium
Explanationg(9) = f(4) + 2 = 16 + 2 = 18.
4
If f(x) = 9x − 18, what is f(-11) − f(8)?
Medium
Explanationf(-11) = -117 and f(8) = 54, so f(-11) − f(8) = -171.
5
A geometric sequence has first term 15 and second term -10. What is the common ratio?
Medium
ExplanationCommon ratio = a₂/a₁ = -10/15 = -2/3.
6
If f(x) = x + 12, what is f(13) − f(6)?
Medium
Explanationf(13) = 25 and f(6) = 18, so f(13) − f(6) = 7.
7
If f(x) = -11x + 6, what is f(-13)?
Medium
Explanationf(-13) = (-11)(-13) + 6 = 149.
8
If f(x) = -2x − 8, for what value of x does f(x) = 22?
Medium
Explanation-2x − 8 = 22, so x = -15.
9
If f(x) = 6x + 2 and g(x) = 5x − 4, what is f(g(10))?
Medium
Explanationg(10) = 46, so f(g(10)) = f(46) = 6(46) + 2 = 278.
10
A population of 975 grows by 20% each year. What will the population be after 2 years?
Medium
ExplanationPopulation after 2 years = 975 × (1 + 20/100)^2 = 1404.
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11
For the function f(x) = 1/(x − 73), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − 73 cannot equal 0, so x = 73 must be excluded from the domain.
12
Is the sequence 9, 7, 5, 3, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term differs from the previous by a constant amount, so this sequence is arithmetic.
13
In an arithmetic sequence, the 5th term is 30 and the 11th term is 84. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (84 − 30)/(11−5) = 9.
14
If f(x) = 6x − 12 and g(x) = -4x + 12, what is f(g(-9))?
Medium
Explanationg(-9) = 48, so f(g(-9)) = f(48) = 6(48) − 12 = 276.
15
A population of 2000 grows by 10% each year. What will the population be after 2 years?
Medium
ExplanationPopulation after 2 years = 2000 × (1 + 10/100)^2 = 2420.
16
What is the sum to infinity of a geometric sequence with first term -14 and common ratio -3/4?
Medium
ExplanationSum to infinity = a₁/(1 − r) = -14/(1 − (-3)/4) = -8.
17
An arithmetic sequence has first term -26 and common difference -9. What is the 14th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -26 + (14−1)(-9) = -143.
18
For the function f(x) = √(x − (-46)), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − (-46) ≥ 0, so x ≥ -46. The smallest such value is -46.
19
What is the sum to infinity of a geometric sequence with first term 9 and common ratio -3/7?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 9/(1 − (-3)/7) = 63/10.
20
An employee's salary starts at Rs. 60058 and increases by Rs. 2045 each year. What will the salary be after 6 years?
Medium
ExplanationSalary after 6 years = 60058 + 6 × 2045 = Rs. 72328.
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21
An arithmetic sequence has first term -13 and common difference -4. What is the 32nd term?
Medium
Explanationaₙ = a₁ + (n−1)d = -13 + (32−1)(-4) = -137.
22
If f(x) = 11x − 5, what is f⁻¹(105)?
Medium
Explanationf⁻¹(105) is the x-value such that f(x) = 105: 11x − 5 = 105, so x = 10.
23
What is the sum to infinity of a geometric sequence with first term 16 and common ratio -1/2?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 16/(1 − (-1)/2) = 32/3.
24
A geometric sequence has first term 5 and common ratio -2. What is the 4th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = 5 × (-2)^3 = -40.
25
If f(x) = -3x + 4 and g(x) = 3x − 14, what is g(f(4))?
Medium
Explanationf(4) = -8, so g(f(4)) = g(-8) = 3(-8) − 14 = -38.
26
A geometric sequence has first term -14 and second term 56. What is the common ratio?
Medium
ExplanationCommon ratio = a₂/a₁ = 56/-14 = -4.
27
What is the sum of the first 8 terms of a geometric sequence with first term -1 and common ratio 3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = -1((3)^8 − 1)/(3 − 1) = -3280.
28
If f(x) = 4x − 11 and g(x) = 2x − 3, what is g(f(-10))?
Medium
Explanationf(-10) = -51, so g(f(-10)) = g(-51) = 2(-51) − 3 = -105.
29
If f(x) = -2x + 5, what is f(3) − f(-10)?
Medium
Explanationf(3) = -1 and f(-10) = 25, so f(3) − f(-10) = -26.
30
If f(x) = 7x − 23, what is f(-11) + f(2)?
Medium
Explanationf(-11) = -100 and f(2) = -9, so f(-11) + f(2) = -109.
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31
If f(x) = 12x + 6, what is f⁻¹(114)?
Medium
Explanationf⁻¹(114) is the x-value such that f(x) = 114: 12x + 6 = 114, so x = 9.
32
If f(x) = 7x + 2, what is f⁻¹(-68)?
Medium
Explanationf⁻¹(-68) is the x-value such that f(x) = -68: 7x + 2 = -68, so x = -10.
33
If f(x) = -x − 8 and g(x) = -7x − 1, what is g(f(10))?
Medium
Explanationf(10) = -18, so g(f(10)) = g(-18) = -7(-18) − 1 = 125.
34
If f(x) = 6x + 3, what is f⁻¹(9)?
Medium
Explanationf⁻¹(9) is the x-value such that f(x) = 9: 6x + 3 = 9, so x = 1.
35
A sequence is defined by a₁ = 18 and aₙ = aₙ₋₁ + 4 for n > 1. What is a5?
Medium
ExplanationEach term adds 4 to the previous one, so a5 = a₁ + (5−1)(4) = 34.
36
In an arithmetic sequence, the 6th term is 95 and the 12th term is 161. What is the 29th term?
Medium
Explanationd = (161 − 95)/(12−6) = 11. Then the 29th term = 95 + (29−6)(11) = 348.
37
In an arithmetic sequence, the 9th term is 70 and the 15th term is 118. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (118 − 70)/(15−9) = 8.
38
An arithmetic sequence has first term -45 and common difference 6. What is the 24th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -45 + (24−1)(6) = 93.
39
If f(x) = 9x + 20, what is f(-4) − f(13)?
Medium
Explanationf(-4) = -16 and f(13) = 137, so f(-4) − f(13) = -153.
40
What is the sum of the first 23 terms of an arithmetic sequence with first term -28 and common difference 3?
Medium
ExplanationThe 23rd term is 38. Sum = n(a₁ + aₙ)/2 = 23(-28 + 38)/2 = 115.
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41
In an arithmetic sequence, the 6th term is 18 and the 16th term is 58. What is the 1st term?
Medium
Explanationd = (58 − 18)/(16−6) = 4. Then the 1st term = 18 + (1−6)(4) = -2.
42
If f(x) = -9x − 9, what is f(-8) + f(8)?
Medium
Explanationf(-8) = 63 and f(8) = -81, so f(-8) + f(8) = -18.
43
What is the sum of the first 8 terms of a geometric sequence with first term -4 and common ratio 2?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = -4((2)^8 − 1)/(2 − 1) = -1020.
44
What is the sum of the first 3 terms of a geometric sequence with first term 2 and common ratio -2?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 2((-2)^3 − 1)/(-2 − 1) = 6.
45
A population of 10240 grows by 25% each year. What will the population be after 5 years?
Medium
ExplanationPopulation after 5 years = 10240 × (1 + 25/100)^5 = 31250.
46
If f(x) = 7x − 7 and g(x) = f(x + 11) + 14, what is g(13)?
Medium
Explanationg(13) = f(24) + 14 = 161 + 14 = 175.
47
If f(x) = 8x − 8 and g(x) = f(x + 9) + 10, what is g(-14)?
Medium
Explanationg(-14) = f(-5) + 10 = -48 + 10 = -38.
48
A population of 4375 grows by 20% each year. What will the population be after 4 years?
Medium
ExplanationPopulation after 4 years = 4375 × (1 + 20/100)^4 = 9072.
49
What is the sum of the first 8 terms of an arithmetic sequence with first term -23 and common difference -6?
Medium
ExplanationThe 8th term is -65. Sum = n(a₁ + aₙ)/2 = 8(-23 + (-65))/2 = -352.
50
If f(x) = 6x − 12, what is f⁻¹(-66)?
Medium
Explanationf⁻¹(-66) is the x-value such that f(x) = -66: 6x − 12 = -66, so x = -9.
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51
What is the sum of the first 14 terms of an arithmetic sequence with first term -4 and common difference -10?
Medium
ExplanationThe 14th term is -134. Sum = n(a₁ + aₙ)/2 = 14(-4 + (-134))/2 = -966.
52
A population of 2000 grows by 10% each year. What will the population be after 3 years?
Medium
ExplanationPopulation after 3 years = 2000 × (1 + 10/100)^3 = 2662.
53
A geometric sequence has first term -7 and common ratio 2. What is the 8th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = -7 × (2)^7 = -896.
54
If f(x) = 4x − 2 and g(x) = -8x − 13, what is f(g(5))?
Medium
Explanationg(5) = -53, so f(g(5)) = f(-53) = 4(-53) − 2 = -214.
55
f(x) = -2x + 13 if x < -5, and f(x) = 6x − 10 if x ≥ -5. What is f(-16)?
Medium
ExplanationSince -16 < -5, use f(x) = -2x + 13: f(-16) = 45.
56
If f(x) = -2x + 3 and g(x) = -5x + 7, what is g(f(-7))?
Medium
Explanationf(-7) = 17, so g(f(-7)) = g(17) = -5(17) + 7 = -78.
57
If f(x) = -x − 24, for what value of x does f(x) = -15?
Medium
Explanation-x − 24 = -15, so x = -9.
58
A sequence is defined by a₁ = 4 and aₙ = aₙ₋₁ + 7 for n > 1. What is a12?
Medium
ExplanationEach term adds 7 to the previous one, so a12 = a₁ + (12−1)(7) = 81.
59
If f(x) = 8x + 12 and g(x) = 8x + 5, what is f(g(0))?
Medium
Explanationg(0) = 5, so f(g(0)) = f(5) = 8(5) + 12 = 52.
60
If f(x) = x, what is f(-2)?
Medium
Explanationf(-2) = (1)(-2) = -2.
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61
If f(x) = 9x − 2, what is f⁻¹(115)?
Medium
Explanationf⁻¹(115) is the x-value such that f(x) = 115: 9x − 2 = 115, so x = 13.
62
A sequence is defined by a₁ = 15 and aₙ = aₙ₋₁ + 4 for n > 1. What is a4?
Medium
ExplanationEach term adds 4 to the previous one, so a4 = a₁ + (4−1)(4) = 27.
63
For the function f(x) = √(x − 43), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − 43 ≥ 0, so x ≥ 43. The smallest such value is 43.
64
Is the sequence 5, 15, 45, 135, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term is a constant multiple of the previous one, so this sequence is geometric.
65
What is the sum of the first 26 terms of an arithmetic sequence with first term -23 and common difference 9?
Medium
ExplanationThe 26th term is 202. Sum = n(a₁ + aₙ)/2 = 26(-23 + 202)/2 = 2327.
66
What is the sum of the first 8 terms of an arithmetic sequence with first term 10 and common difference -7?
Medium
ExplanationThe 8th term is -39. Sum = n(a₁ + aₙ)/2 = 8(10 + (-39))/2 = -116.
67
A geometric sequence has first term 9 and common ratio 5. What is the 6th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = 9 × (5)^5 = 28125.
68
A sequence is defined by a₁ = 5 and aₙ = aₙ₋₁ − 9 for n > 1. What is a6?
Medium
ExplanationEach term adds -9 to the previous one, so a6 = a₁ + (6−1)(-9) = -40.
69
f(x) = -5x − 7 if x < -8, and f(x) = -4x − 6 if x ≥ -8. What is f(-19)?
Medium
ExplanationSince -19 < -8, use f(x) = -5x − 7: f(-19) = 88.
70
If f(x) = 4x + 7 and g(x) = -8x + 14, what is g(f(8))?
Medium
Explanationf(8) = 39, so g(f(8)) = g(39) = -8(39) + 14 = -298.
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71
If f(x) = -3x^2 − 13x − 21, what is f(-10)?
Medium
Explanationf(-10) = -3(-10)^2 + (-13)(-10) + (-21) = -191.
72
If f(x) = 8x + 2, what is f⁻¹(-38)?
Medium
Explanationf⁻¹(-38) is the x-value such that f(x) = -38: 8x + 2 = -38, so x = -5.
73
If f(x) = -7x + 6 and g(x) = 8x + 6, what is g(f(-8))?
Medium
Explanationf(-8) = 62, so g(f(-8)) = g(62) = 8(62) + 6 = 502.
74
What is the sum of the first 3 terms of a geometric sequence with first term -6 and common ratio 4?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = -6((4)^3 − 1)/(4 − 1) = -126.
75
If f(x) = 4x^2 + 2x − 3, what is f(-8)?
Medium
Explanationf(-8) = 4(-8)^2 + (2)(-8) + (-3) = 237.
76
What is the sum of the first 3 terms of a geometric sequence with first term 5 and common ratio -3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 5((-3)^3 − 1)/(-3 − 1) = 35.
77
A population of 1472 grows by 25% each year. What will the population be after 3 years?
Medium
ExplanationPopulation after 3 years = 1472 × (1 + 25/100)^3 = 2875.
78
If f(x) = 6x^2 − 8x + 3, what is f(6)?
Medium
Explanationf(6) = 6(6)^2 + (-8)(6) + (3) = 171.
79
A sequence is defined by a₁ = 2 and aₙ = aₙ₋₁ − 7 for n > 1. What is a7?
Medium
ExplanationEach term adds -7 to the previous one, so a7 = a₁ + (7−1)(-7) = -40.
80
In an arithmetic sequence, the 5th term is -28 and the 15th term is -68. What is the 24th term?
Medium
Explanationd = (-68 − (-28))/(15−5) = -4. Then the 24th term = -28 + (24−5)(-4) = -104.
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81
Is the sequence -10, 30, -90, 270, ... arithmetic, geometric, or neither?
Medium
ExplanationEach term is a constant multiple of the previous one, so this sequence is geometric.
82
If f(x) = -5x + 4, for what value of x does f(x) = 59?
Medium
Explanation-5x + 4 = 59, so x = -11.
83
What is the sum to infinity of a geometric sequence with first term 15 and common ratio -1/4?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 15/(1 − (-1)/4) = 12.
84
If f(x) = 4x + 10, what is f(-3) + f(13)?
Medium
Explanationf(-3) = -2 and f(13) = 62, so f(-3) + f(13) = 60.
85
If f(x) = 3x − 19, for what value of x does f(x) = 2?
Medium
Explanation3x − 19 = 2, so x = 7.
86
For the function f(x) = 1/(x − 7), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − 7 cannot equal 0, so x = 7 must be excluded from the domain.
87
Is the sequence 15, 7, 10, 0, ... arithmetic, geometric, or neither?
Medium
ExplanationThere is no constant common difference or common ratio between terms, so this sequence is neither arithmetic nor geometric.
88
In an arithmetic sequence, the 13th term is -42 and the 17th term is -70. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-70 − (-42))/(17−13) = -7.
89
An employee's salary starts at Rs. 67782 and increases by Rs. 3223 each year. What will the salary be after 10 years?
Medium
ExplanationSalary after 10 years = 67782 + 10 × 3223 = Rs. 100012.
90
For the function f(x) = 1/(x − (-45)), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − (-45) cannot equal 0, so x = -45 must be excluded from the domain.
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91
If f(x) = x + 4 and g(x) = -4x − 2, what is f(g(-3))?
Medium
Explanationg(-3) = 10, so f(g(-3)) = f(10) = 1(10) + 4 = 14.
92
What is the sum of the first 8 terms of a geometric sequence with first term 4 and common ratio -3?
Medium
ExplanationSum = a₁(rⁿ − 1)/(r − 1) = 4((-3)^8 − 1)/(-3 − 1) = -6560.
93
If f(x) = 8x + 14 and g(x) = 8x + 8, what is g(f(-9))?
Medium
Explanationf(-9) = -58, so g(f(-9)) = g(-58) = 8(-58) + 8 = -456.
94
In an arithmetic sequence, the 3rd term is 17 and the 12th term is 26. What is the 20th term?
Medium
Explanationd = (26 − 17)/(12−3) = 1. Then the 20th term = 17 + (20−3)(1) = 34.
95
What is the sum to infinity of a geometric sequence with first term 6 and common ratio 4/5?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 6/(1 − 4/5) = 30.
96
If f(x) = x + 21, for what value of x does f(x) = 17?
Medium
Explanationx + 21 = 17, so x = -4.
97
If f(x) = 4x − 18, what is f⁻¹(42)?
Medium
Explanationf⁻¹(42) is the x-value such that f(x) = 42: 4x − 18 = 42, so x = 15.
98
If f(x) = 8x + 5 and g(x) = -6x + 5, what is g(f(0))?
Medium
Explanationf(0) = 5, so g(f(0)) = g(5) = -6(5) + 5 = -25.
99
For the function f(x) = √(x − (-35)), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − (-35) ≥ 0, so x ≥ -35. The smallest such value is -35.
100
An employee's salary starts at Rs. 20379 and increases by Rs. 1267 each year. What will the salary be after 11 years?
Medium
ExplanationSalary after 11 years = 20379 + 11 × 1267 = Rs. 34316.