Relations & Functions – Domain, Range, Types of Functions Quiz - ICSE Class 11 Mathematics

Free practice quiz: Relations & Functions – Domain, Range, Types of Functions (ICSE Class 11 Mathematics). 54 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 54 multiple-choice questions on Relations & Functions – Domain, Range, Types of Functions, and 54 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. Let f(x) = \frac{1}{\sqrt{x^2 - 4}}. Find the domain of f(x).

   1. (-∞, -2) ∪ (2, ∞)
   2. (-∞, -2] ∪ [2, ∞)
   3. (-2, 2)
   4. [-2, 2]
   Show answer

   Answer: A. (-∞, -2) ∪ (2, ∞)

   For f(x) to be defined, the expression under the square root must be positive, i.e., $x^2 - 4 > 0$. This inequality holds when $x^2 > 4$, which means $x < -2$ or $x > 2$. Therefore, the domain is (-∞, -2) ∪ (2, ∞).

2. Let $f(x) = |x-3|$. What is the range of $f(x)$?

   1. [0, ∞)
   2. (0, ∞)
   3. ℝ
   4. [3, ∞)
   Show answer

   Answer: A. [0, ∞)

   The absolute value of any real number is always non-negative. The expression $|x-3|$ will be zero when $x=3$. For all other values of $x$, $|x-3|$ will be positive. Therefore, the range of $f(x) = |x-3|$ is all non-negative real numbers, which is the interval [0, ∞).

3. Determine the range of $f(x) = |x| + 2$.

   1. [2, ∞)
   2. (2, ∞)
   3. [0, ∞)
   4. ℝ
   Show answer

   Answer: A. [2, ∞)

   The term $|x|$ is always non-negative, meaning $|x| \ge 0$. Adding 2 to $|x|$, we get $|x| + 2 \ge 0 + 2$, so $f(x) \ge 2$. The minimum value of $f(x)$ is 2, which occurs when $x=0$. As $|x|$ increases, $f(x)$ increases without bound. Thus, the range is [2, ∞).

4. Let $f(x) = \frac{x}{x^2-9}$. Find the domain of $f(x)$.

   1. ℝ \{-3, 3}
   2. ℝ \{3}
   3. ℝ \{-3}
   4. ℝ \{0, 3, -3}
   Show answer

   Answer: A. ℝ \{-3, 3}

   The function $f(x) = \frac{x}{x^2-9}$ is a rational function. It is defined for all real numbers except where the denominator is zero. The denominator $x^2-9 = (x-3)(x+3)$ is zero when $x=3$ or $x=-3$. Therefore, the domain of $f(x)$ is all real numbers except 3 and -3.

5. If $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x}$, find the domain of $(f \circ g)(x)$.

   1. ℝ \{0}
   2. ℝ \{1}
   3. ℝ
   4. ℝ \{-1, 0, 1}
   Show answer

   Answer: A. ℝ \{0}

   The composite function $(f \circ g)(x) = f(g(x))$. The domain of $g(x) = \frac{1}{x}$ is ℝ \{0}. Now we evaluate $f(g(x)) = f(\frac{1}{x}) = \frac{1}{\frac{1}{x}} = x$. The resulting expression is $x$, whose domain is ℝ. However, the domain of the composite function is restricted by the domain of the inner function $g(x)$. Thus, the domain of $(f \circ g)(x)$ is ℝ \{0}.

More questions in this quiz

Answers and explanations are in the full quiz.

1. If $f(x) = \sqrt{9 - x^2}$, find the range of f(x).
2. Which of the following is an odd function?
3. Let $f(x) = \frac{x-1}{x+1}$. Find the inverse function $f^{-1}(x)$.
4. If $f(x) = \lfloor x \rfloor$ (greatest integer function), find the range of f(x).
5. Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x + 2$. Is this function one-to-one?
6. Let $f(x) = \frac{1}{x-2}$. What is the domain of $f(x)$?
7. Determine the range of the function $f(x) = x^2 + 1$.
8. Which of the following functions is neither even nor odd?
9. If $f(x) = \log_a(x)$, where $a > 0$ and $a \ne 1$, find the domain of $f(x)$.
10. If $f(x) = \frac{x^2 - 4}{x - 2}$, what is the domain of $f(x)$?
11. Consider the function $f(x) = x^2$. Is this function onto (surjective)?
12. Let $f(x) = e^x$. Which statement is true about this function?
13. Find the domain of the function $g(x) = \sqrt{x-5} + \sqrt{10-x}$.
14. If $f(x) = x^2 + 2x + 1$, what is the range of $f(x)$?
15. Which of the following functions is a bijection (one-to-one and onto)?
16. Let $f(x) = \tan(x)$. Find the domain of $f(x)$.
17. Determine the range of $f(x) = \sin(x)$.
18. If $f(x) = x^3$, is this function one-to-one?
19. Let $f(x) = \frac{1}{\sqrt{x-1}}$. Find the domain of $f(x)$.
20. If $f(x) = \frac{1}{x}$, which of the following is true?
21. Let $f(x) = \sqrt{16 - x^2}$. Find the domain of $f(x)$.
22. What is the range of the function $f(x) = \cos(x)$?
23. Consider $f(x) = x^2 - 2x + 3$. Find the range of $f(x)$.
24. Which of the following is a function that is neither one-to-one nor onto from ℝ to ℝ?
25. Let $f(x) = \frac{x+1}{x-1}$. Find the range of $f(x)$.
26. Which of the following is an even function?
27. Let $f(x) = \frac{1}{\sqrt{1-x^2}}$. Find the domain of $f(x)$.
28. Determine the range of $f(x) = 2^x$.
29. If $f(x) = x^2$ and $g(x) = \sqrt{x}$, find the domain of the composite function $(f \circ g)(x)$.
30. Which of the following is a function that is both one-to-one and onto from ℝ to ℝ?
31. Let $f(x) = \sin^{-1}(x)$. Find the domain of $f(x)$.
32. Determine the range of $f(x) = \sqrt{x^2 + 1}$.
33. If $f(x) = x^2 + 2x + 1$ and $g(x) = x - 1$, find the domain of $(g \circ f)(x)$.
34. Let $f(x) = \frac{1}{x^2-4}$. Find the domain of $f(x)$.
35. Which of the following functions is an odd function?
36. Let $f(x) = \ln(x)$. Find the range of $f(x)$.
37. If $f(x) = \frac{\sqrt{x-2}}{x-5}$, find the domain of $f(x)$.
38. Let $f(x) = \frac{1}{\sqrt{4-x^2}}$. Find the domain of $f(x)$.
39. Determine the range of $f(x) = e^{-x^2}$.
40. Which of the following is a one-to-one function?
41. Let $f(x) = \frac{1}{\sqrt{x-5}}$. Find the domain of $f(x)$.
42. Determine the range of $f(x) = \sec(x)$.
43. Let $f(x) = \sqrt{x^2-9}$. Find the domain of $f(x)$.
44. Determine the range of $f(x) = 3^x$.
45. If $f(x) = x^2$ and $g(x) = x+1$, find the domain of $(f \circ g)(x)$.
46. Let $f(x) = \sqrt{x^2 - 16}$. Find the domain of $f(x)$.
47. Determine the range of $f(x) = |x-2| + 3$.
48. If $f(x) = \frac{1}{x-1}$ and $g(x) = x+1$, find the domain of $(g \circ f)(x)$.
49. Let $f(x) = \frac{1}{x^2}$. Find the domain of $f(x)$.

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