TGT PGT Mathematics: Inverse Functions Objective Questions
Year: 2025
Score: 0/50
1. For a function f to have an inverse, it must be:
Correct Answer: c) One-to-one and onto
Explanation: A function has an inverse if it is bijective (one-to-one and onto).
Explanation: A function has an inverse if it is bijective (one-to-one and onto).
2. The inverse of f(x) = 2x + 3 is:
Correct Answer: a) (x-3)/2
Explanation: Solve y = 2x + 3 for x: x = (y-3)/2, so f⁻¹(x) = (x-3)/2.
Explanation: Solve y = 2x + 3 for x: x = (y-3)/2, so f⁻¹(x) = (x-3)/2.
3. Which function does not have an inverse over all real numbers?
Correct Answer: c) x²
Explanation: f(x) = x² is not one-to-one (e.g., f(2) = f(-2)), so it has no inverse over all reals.
Explanation: f(x) = x² is not one-to-one (e.g., f(2) = f(-2)), so it has no inverse over all reals.
4. The inverse of f(x) = ln x is:
Correct Answer: a) e^x
Explanation: If y = ln x, then x = e^y, so f⁻¹(x) = e^x.
Explanation: If y = ln x, then x = e^y, so f⁻¹(x) = e^x.
5. If f(x) = x³ + 1, then f⁻¹(9) is:
Correct Answer: a) 2
Explanation: Solve f(x) = 9: x³ + 1 = 9 → x³ = 8 → x = 2, so f⁻¹(9) = 2.
Explanation: Solve f(x) = 9: x³ + 1 = 9 → x³ = 8 → x = 2, so f⁻¹(9) = 2.
6. The domain of f⁻¹ for f(x) = 1/(x-2) is:
Correct Answer: b) x ≠ 0
Explanation: Range of f(x) is x ≠ 0, which is the domain of f⁻¹.
Explanation: Range of f(x) is x ≠ 0, which is the domain of f⁻¹.
7. If f(x) = sin x, [-π/2, π/2], then f⁻¹(x) is:
Correct Answer: b) arcsin x
Explanation: On [-π/2, π/2], sin x is one-to-one, so f⁻¹(x) = arcsin x.
Explanation: On [-π/2, π/2], sin x is one-to-one, so f⁻¹(x) = arcsin x.
8. The graph of f⁻¹ is obtained by reflecting f over:
Correct Answer: c) Line y = x
Explanation: The graph of f⁻¹ is the reflection of f over y = x.
Explanation: The graph of f⁻¹ is the reflection of f over y = x.
9. If f(x) = e^x, then f⁻¹(x) is:
Correct Answer: a) ln x
Explanation: If y = e^x, then x = ln y, so f⁻¹(x) = ln x.
Explanation: If y = e^x, then x = ln y, so f⁻¹(x) = ln x.
10. The inverse of f(x) = x/(x+1), x ≠ -1, is:
Correct Answer: a) x/(1-x)
Explanation: Solve y = x/(x+1): x = y/(1-y), so f⁻¹(x) = x/(1-x).
Explanation: Solve y = x/(x+1): x = y/(1-y), so f⁻¹(x) = x/(1-x).
11. If f(x) = x², x ≥ 0, then f⁻¹(x) is:
Correct Answer: a) √x
Explanation: For x ≥ 0, y = x² → x = √y, so f⁻¹(x) = √x.
Explanation: For x ≥ 0, y = x² → x = √y, so f⁻¹(x) = √x.
12. The derivative of f⁻¹(x) at x = a is:
Correct Answer: c) 1/f'(f⁻¹(a))
Explanation: By the inverse function theorem, (f⁻¹)'(x) = 1/f'(f⁻¹(x)).
Explanation: By the inverse function theorem, (f⁻¹)'(x) = 1/f'(f⁻¹(x)).
13. If f(x) = tan x, [-π/4, π/4], then f⁻¹(x) is:
Correct Answer: a) arctan x
Explanation: On [-π/4, π/4], tan x is one-to-one, so f⁻¹(x) = arctan x.
Explanation: On [-π/4, π/4], tan x is one-to-one, so f⁻¹(x) = arctan x.
14. The inverse of f(x) = 3x – 2 is:
Correct Answer: a) (x+2)/3
Explanation: Solve y = 3x – 2: x = (y+2)/3, so f⁻¹(x) = (x+2)/3.
Explanation: Solve y = 3x – 2: x = (y+2)/3, so f⁻¹(x) = (x+2)/3.
15. If f(x) = x³ – x, does f have an inverse over all real numbers?
Correct Answer: b) No
Explanation: f(x) = x³ – x is not one-to-one (e.g., f(1) = f(-1)), so no inverse over all reals.
Explanation: f(x) = x³ – x is not one-to-one (e.g., f(1) = f(-1)), so no inverse over all reals.
16. The range of f⁻¹ for f(x) = x² + 2, x ≥ 0, is:
Correct Answer: a) y ≥ 0
Explanation: Domain of f is x ≥ 0, so range of f⁻¹ is y ≥ 0.
Explanation: Domain of f is x ≥ 0, so range of f⁻¹ is y ≥ 0.
17. If f(x) = arcsin x, then f(x) is:
Correct Answer: b) The inverse of sin x on [-π/2, π/2]
Explanation: arcsin x is the inverse of sin x restricted to [-π/2, π/2].
Explanation: arcsin x is the inverse of sin x restricted to [-π/2, π/2].
18. The inverse of f(x) = 5x + 4 is:
Correct Answer: a) (x-4)/5
Explanation: Solve y = 5x + 4: x = (y-4)/5, so f⁻¹(x) = (x-4)/5.
Explanation: Solve y = 5x + 4: x = (y-4)/5, so f⁻¹(x) = (x-4)/5.
19. If f(x) = 2^x, then f⁻¹(x) is:
Correct Answer: a) log₂ x
Explanation: If y = 2^x, then x = log₂ y, so f⁻¹(x) = log₂ x.
Explanation: If y = 2^x, then x = log₂ y, so f⁻¹(x) = log₂ x.
20. If f(x) = x + 1/x, x ≠ 0, does f have an inverse over all reals except 0?
Correct Answer: b) No
Explanation: f(x) = x + 1/x is not one-to-one (e.g., f(1) = f(-1)), so no inverse over x ≠ 0.
Explanation: f(x) = x + 1/x is not one-to-one (e.g., f(1) = f(-1)), so no inverse over x ≠ 0.
21. The domain of f⁻¹ for f(x) = cos x, [0, π], is:
Correct Answer: a) [-1, 1]
Explanation: Range of f(x) = cos x on [0, π] is [-1, 1], which is the domain of f⁻¹.
Explanation: Range of f(x) = cos x on [0, π] is [-1, 1], which is the domain of f⁻¹.
22. If f(x) = x² – 4x + 3, x ≥ 2, then f⁻¹(x) is:
Correct Answer: a) 2 + √(x+1)
Explanation: Solve y = (x-2)² – 1: x = 2 ± √(y+1). Since x ≥ 2, f⁻¹(x) = 2 + √(x+1).
Explanation: Solve y = (x-2)² – 1: x = 2 ± √(y+1). Since x ≥ 2, f⁻¹(x) = 2 + √(x+1).
23. If f(x) = 10^x, then f⁻¹(x) is:
Correct Answer: a) log₁₀ x
Explanation: If y = 10^x, then x = log₁₀ y, so f⁻¹(x) = log₁₀ x.
Explanation: If y = 10^x, then x = log₁₀ y, so f⁻¹(x) = log₁₀ x.
24. If f(x) = x³ + 2, then f⁻¹(10) is:
Correct Answer: d) ∛8
Explanation: Solve f(x) = 10: x³ + 2 = 10 → x³ = 8 → x = ∛8, so f⁻¹(10) = ∛8.
Explanation: Solve f(x) = 10: x³ + 2 = 10 → x³ = 8 → x = ∛8, so f⁻¹(10) = ∛8.
25. The inverse of f(x) = (x-1)/(x+1), x ≠ -1, is:
Correct Answer: a) (x+1)/(1-x)
Explanation: Solve y = (x-1)/(x+1): x = (y+1)/(1-y), so f⁻¹(x) = (x+1)/(1-x).
Explanation: Solve y = (x-1)/(x+1): x = (y+1)/(1-y), so f⁻¹(x) = (x+1)/(1-x).
26. If f(x) = arctan x, then f(x) is:
Correct Answer: b) The inverse of tan x on [-π/2, π/2]
Explanation: arctan x is the inverse of tan x restricted to [-π/2, π/2].
Explanation: arctan x is the inverse of tan x restricted to [-π/2, π/2].
27. If f(x) = x², x ≤ 0, then f⁻¹(x) is:
Correct Answer: b) -√x
Explanation: For x ≤ 0, y = x² → x = -√y, so f⁻¹(x) = -√x.
Explanation: For x ≤ 0, y = x² → x = -√y, so f⁻¹(x) = -√x.
28. The inverse of f(x) = 4x + 5 is:
Correct Answer: a) (x-5)/4
Explanation: Solve y = 4x + 5: x = (y-5)/4, so f⁻¹(x) = (x-5)/4.
Explanation: Solve y = 4x + 5: x = (y-5)/4, so f⁻¹(x) = (x-5)/4.
29. If f(x) = sinh x, then f⁻¹(x) is:
Correct Answer: a) arcsinh x
Explanation: The inverse of sinh x is arcsinh x.
Explanation: The inverse of sinh x is arcsinh x.
30. If f(x) = x³ – 2x, does f have an inverse over all reals?
Correct Answer: b) No
Explanation: f(x) = x³ – 2x is not one-to-one (e.g., f(1) = f(-1)), so no inverse over all reals.
Explanation: f(x) = x³ – 2x is not one-to-one (e.g., f(1) = f(-1)), so no inverse over all reals.
31. The range of f⁻¹ for f(x) = x/(x-1), x ≠ 1, is:
Correct Answer: a) x ≠ 1
Explanation: Domain of f is x ≠ 1, so range of f⁻¹ is x ≠ 1.
Explanation: Domain of f is x ≠ 1, so range of f⁻¹ is x ≠ 1.
32. If f(x) = arccos x, then f(x) is:
Correct Answer: b) The inverse of cos x on [0, π]
Explanation: arccos x is the inverse of cos x restricted to [0, π].
Explanation: arccos x is the inverse of cos x restricted to [0, π].
33. The inverse of f(x) = 2x², x ≥ 0, is:
Correct Answer: a) √(x/2)
Explanation: Solve y = 2x²: x = √(y/2), so f⁻¹(x) = √(x/2).
Explanation: Solve y = 2x²: x = √(y/2), so f⁻¹(x) = √(x/2).
34. If f(x) = x³ + 3x, does f have an inverse over all reals?
Correct Answer: a) Yes
Explanation: f'(x) = 3x² + 3 > 0, so f is strictly increasing and has an inverse over all reals.
Explanation: f'(x) = 3x² + 3 > 0, so f is strictly increasing and has an inverse over all reals.
35. The inverse of f(x) = 1/x, x ≠ 0, is:
Correct Answer: a) 1/x
Explanation: Solve y = 1/x: x = 1/y, so f⁻¹(x) = 1/x.
Explanation: Solve y = 1/x: x = 1/y, so f⁻¹(x) = 1/x.
36. If f(x) = ln(x+1), x > -1, then f⁻¹(x) is:
Correct Answer: a) e^x – 1
Explanation: Solve y = ln(x+1): x = e^y – 1, so f⁻¹(x) = e^x – 1.
Explanation: Solve y = ln(x+1): x = e^y – 1, so f⁻¹(x) = e^x – 1.
37. The domain of f⁻¹ for f(x) = x² – 2x + 1, x ≥ 1, is:
Correct Answer: a) y ≥ 0
Explanation: f(x) = (x-1)², x ≥ 1, has range y ≥ 0, which is the domain of f⁻¹.
Explanation: f(x) = (x-1)², x ≥ 1, has range y ≥ 0, which is the domain of f⁻¹.
38. If f(x) = arcsec x, then the domain of f is:
Correct Answer: b) |x| ≥ 1
Explanation: arcsec x is defined for |x| ≥ 1.
Explanation: arcsec x is defined for |x| ≥ 1.
39. The inverse of f(x) = x³ + x is:
Correct Answer: a) Not expressible in simple terms
Explanation: f(x) = x³ + x is one-to-one, but its inverse is not easily expressible algebraically.
Explanation: f(x) = x³ + x is one-to-one, but its inverse is not easily expressible algebraically.
40. If f(x) = x² + x, x ≥ -1/2, then f⁻¹(2) is:
Correct Answer: a) 1
Explanation: Solve x² + x = 2: x² + x – 2 = 0 → x = 1 or x = -2. Since x ≥ -1/2, x = 1, so f⁻¹(2) = 1.
Explanation: Solve x² + x = 2: x² + x – 2 = 0 → x = 1 or x = -2. Since x ≥ -1/2, x = 1, so f⁻¹(2) = 1.
41. The inverse of f(x) = 3^x is:
Correct Answer: a) log₃ x
Explanation: If y = 3^x, then x = log₃ y, so f⁻¹(x) = log₃ x.
Explanation: If y = 3^x, then x = log₃ y, so f⁻¹(x) = log₃ x.
42. If f(x) = x/(x+2), x ≠ -2, then f⁻¹(x) is:
Correct Answer: a) 2x/(1-x)
Explanation: Solve y = x/(x+2): x = 2y/(1-y), so f⁻¹(x) = 2x/(1-x).
Explanation: Solve y = x/(x+2): x = 2y/(1-y), so f⁻¹(x) = 2x/(1-x).
43. The range of f⁻¹ for f(x) = sin x, [-π/2, π/2], is:
Correct Answer: b) [-π/2, π/2]
Explanation: Domain of f is [-π/2, π/2], so range of f⁻¹ is [-π/2, π/2].
Explanation: Domain of f is [-π/2, π/2], so range of f⁻¹ is [-π/2, π/2].
44. If f(x) = x² + 2x + 1, x ≥ -1, then f⁻¹(x) is:
Correct Answer: b) -1 + √x
Explanation: f(x) = (x+1)², x ≥ -1. Solve y = (x+1)²: x = -1 ± √y. Since x ≥ -1, f⁻¹(x) = -1 + √x.
Explanation: f(x) = (x+1)², x ≥ -1. Solve y = (x+1)²: x = -1 ± √y. Since x ≥ -1, f⁻¹(x) = -1 + √x.
45. If f(x) = e^(2x), then f⁻¹(x) is:
Correct Answer: a) ln x / 2
Explanation: Solve y = e^(2x): x = ln y / 2, so f⁻¹(x) = ln x / 2.
Explanation: Solve y = e^(2x): x = ln y / 2, so f⁻¹(x) = ln x / 2.
46. The inverse of f(x) = x/(2-x), x ≠ 2, is:
Correct Answer: a) 2x/(x+1)
Explanation: Solve y = x/(2-x): x = 2y/(y+1), so f⁻¹(x) = 2x/(x+1).
Explanation: Solve y = x/(2-x): x = 2y/(y+1), so f⁻¹(x) = 2x/(x+1).
47. If f(x) = x² – 2, x ≥ 0, then f⁻¹(7) is:
Correct Answer: a) 3
Explanation: Solve x² – 2 = 7: x² = 9 → x = 3 (since x ≥ 0), so f⁻¹(7) = 3.
Explanation: Solve x² – 2 = 7: x² = 9 → x = 3 (since x ≥ 0), so f⁻¹(7) = 3.
48. If f(x) = arccot x, then the domain of f is:
Correct Answer: a) All reals
Explanation: arccot x is defined for all real x.
Explanation: arccot x is defined for all real x.
49. The inverse of f(x) = x³ – 1 is:
Correct Answer: a) ∛(x+1)
Explanation: Solve y = x³ – 1: x = ∛(y+1), so f⁻¹(x) = ∛(x+1).
Explanation: Solve y = x³ – 1: x = ∛(y+1), so f⁻¹(x) = ∛(x+1).
50. If f(x) = x² + 3x + 2, x ≥ -3/2, then f⁻¹(x) is:
Correct Answer: a) -3/2 + √(x+1/4)
Explanation: f(x) = (x+3/2)² – 1/4. Solve y = (x+3/2)² – 1/4: x = -3/2 ± √(y+1/4). Since x ≥ -3/2, f⁻¹(x) = -3/2 + √(x+1/4).
Explanation: f(x) = (x+3/2)² – 1/4. Solve y = (x+3/2)² – 1/4: x = -3/2 ± √(y+1/4). Since x ≥ -3/2, f⁻¹(x) = -3/2 + √(x+1/4).
