Sample Quizzes For Preparation: Functions
1. Which of the following best defines a function?
A. A relationship where every input maps to multiple outputs
B. A rule assigning at least one output to every input
C. A rule that assigns exactly one output for each input
D. Any relationship between x and y
2. What is the domain of the function f(x) = 1/(x – 4)?
A. All real numbers
B. All real numbers except x = 0
C. All real numbers except x = 4
D. x ≥ 0
3. Which function is NOT one–one?
A. f(x) = x + 1
B. f(x) = 2x – 5
C. f(x) = x²
D. f(x) = 3x
4. What is the inverse of the function f(x) = 3x – 7?
A. f⁻¹(x) = (x + 7)/3
B. f⁻¹(x) = (x – 7)/3
C. f⁻¹(x) = 3x + 7
D. f⁻¹(x) = x/3 – 7
5. What must be true for a function to have an inverse?
A. It must be quadratic
B. It must be many–one
C. It must be one–one
D. It must be periodic
6. What is the range of f(x) = √x?
A. x ∈ ℝ
B. x ≥ 0
C. x ≤ 0
D. x ≠ 0
7. Which pair shows correct function and its inverse?
A. f(x) = x², f⁻¹(x) = √x
B. f(x) = x + 3, f⁻¹(x) = x – 3
C. f(x) = 2x + 5, f⁻¹(x) = (x + 5)/2
D. f(x) = log x, f⁻¹(x) = 10x
8. What is the result of the composite function fg(x), if f(x) = 2x and g(x) = x + 3?
A. 2x + 3
B. 2x + 6
C. 2(x + 3)
D. (2x) + 3
9. Which graph transformation is correct for y = |f(x)|?
A. Reflects negative y-values over x-axis
B. Reflects over y-axis
C. Stretches vertically
D. No change to graph
10. Which of the following correctly shows that fg ≠ gf?
A. f(x) = 2x, g(x) = x + 1
B. f(x) = x², g(x) = x + 1
C. f(x) = x – 3, g(x) = x + 3
D. f(x) = x, g(x) = x
11. What is the domain of f⁻¹(x) if f(x) = √(x – 2)?
A. x ≥ 2
B. x ≤ 2
C. x > 0
D. All x
12. Which graph reflects across y = x to produce its inverse?
A. y = |x|
B. y = x + 2
C. y = x²
D. All one–one functions
13. What is f²(x) if f(x) = x + 1?
A. x + 2
B. x + 3
C. f(f(x)) = x + 2
D. x² + 1
14. Which is NOT a valid function notation?
A. f(x)
B. f⁻¹(x)
C. f(f(x))
D. f + x
15. For which value is f(x) = 1/(x – 3) undefined?
A. x = 0
B. x = –3
C. x = 3
D. x = 1
16. What is the domain of the function f(x) = √(5 – x)?
A. x ≥ 5
B. x ≤ 5
C. x > 0
D. All real numbers
17. If f(x) = e^x, then f⁻¹(x) = ?
A. log x
B. ln x
C. e^x
D. x²
18. Which function is many–one?
A. f(x) = x
B. f(x) = √x
C. f(x) = x²
D. f(x) = ln x
19. If f(x) = 2x + 1, what is f⁻¹(7)?
A. 3
B. 2
C. 1
D. 4
20. If f(x) = x – 5 and g(x) = x², find gf(x)
A. x² – 10x + 25
B. (x – 5)²
C. x² – 25
D. x² – 5x
Answer Key and Explanations
1. C – A function assigns exactly one output for each input.
A is incorrect because it allows multiple outputs. B is vague. D is too general.
2. C – Denominator cannot be zero, so x ≠ 4.
A is incorrect because it includes x = 4. B and D are irrelevant.
3. C – f(x) = x² is many–one (e.g., f(2) = f(–2)).
All others are one–one.
4. B – Solve y = 3x – 7 → x = (y + 7)/3 ⇒ f⁻¹(x) = (x + 7)/3
B is correct. A is wrong because it adds before dividing. C and D reverse the function incorrectly.
5. C – Only one–one functions have inverses.
Many–one functions map different inputs to same output.
6. B – Square root function only gives non-negative values (x ≥ 0).
C and D violate this. A is too broad.
7. B – Only f(x) = x + 3 and its inverse f⁻¹(x) = x – 3 are correct.
A is wrong because x² is not one–one. C wrongly adds 5. D confuses inverse of log with 10^x, not 10x.
8. C – fg(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6
A and B are incorrect simplifications. D is redundant.
9. A – y = |f(x)| reflects negative values above x-axis.
Not a stretch or y-axis reflection.
10. A – f(g(x)) = 2(x + 1) = 2x + 2 ≠ g(f(x)) = 2x + 1
Demonstrates fg ≠ gf.
11. A – f(x) = √(x – 2) → domain x ≥ 2, so inverse also x ≥ 0
B, C, D are false domains.
12. D – All one–one functions reflect across y = x to form their inverse.
C is not one–one; A is one–many.
13. C – f(f(x)) = f(x + 1) = (x + 1) + 1 = x + 2
A is incorrect unless asking for just one application.
14. D – f + x is not valid notation.
All others are standard function forms.
15. C – Division by zero at x = 3 makes it undefined.
Others do not result in zero denominator.
16. B – √(5 – x) is only real when (5 – x) ≥ 0 → x ≤ 5
A is reversed. C and D are too general.
17. B – ln x is the inverse of e^x
A (log x) is base 10. C is the original function. D is unrelated.
18. C – x² is many–one (same output for +x and –x)
All others are one–one.
19. A – f(x) = 2x + 1 ⇒ f⁻¹(x) = (x – 1)/2 → f⁻¹(7) = (7 – 1)/2 = 3
B, C, D are miscalculations.
20. B – gf(x) = g(f(x)) = g(x – 5) = (x – 5)²
A is expansion of B. C and D are incorrect compositions.