Relations and Functions Quiz - ICSE Class 12 Mathematics

Free practice quiz: Relations and Functions (ICSE Class 12 Mathematics). 57 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 12 Mathematics practice quiz has 57 multiple-choice questions on Relations and Functions, and 57 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 $f: \mathbb{R} \setminus \{0\} o \mathbb{R}$ be defined by $f(x) = \frac{1}{x}$. If $g(x) = f(f(x))$, find the domain of $g(x)$.

   1. $\mathbb{R}$
   2. $\mathbb{R} \setminus \{0\}$
   3. $\mathbb{R} \setminus \{1\}$
   4. $\mathbb{R} \setminus \{-1\}$
   Show answer

   Answer: B. $\mathbb{R} \setminus \{0\}$

   The function $f(x) = \frac{1}{x}$ has a domain of $\mathbb{R} \setminus \{0\}$. For $g(x) = f(f(x))$, the input to the outer function $f$ is $f(x)$. Therefore, $f(x)$ cannot be $0$. Since $f(x) = \frac{1}{x}$, $\frac{1}{x} eq 0$ is always true for any $x \in \mathbb{R} \setminus \{0\}$. The domain of $f(x)$ is $\mathbb{R} \setminus \{0\}$. Thus, the domain of $g(x)$ is also $\mathbb{R} \setminus \{0\}$.

2. Let $f(x) = 2x + 3$ and $g(x) = x-1$. Find $(f+g)(x)$.

   1. $3x + 2$
   2. $2x^2 + 3x - 1$
   3. $2x - 2$
   4. $3x + 4$
   Show answer

   Answer: A. $3x + 2$

   The sum of two functions $f(x)$ and $g(x)$, denoted by $(f+g)(x)$, is defined as the function obtained by adding the values of $f(x)$ and $g(x)$ for each $x$ in their common domain. In this case, the domain for both $f(x)$ and $g(x)$ is all real numbers ($\mathbb{R}$). $(f+g)(x) = f(x) + g(x) = (2x + 3) + (x - 1)$. Combine like terms: $(f+g)(x) = (2x + x) + (3 - 1) = 3x + 2$.

3. If $f(x) = x^3 - x$, then $f$ is:

   1. Even
   2. Odd
   3. Neither even nor odd
   4. Both even and odd
   Show answer

   Answer: B. Odd

   A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain. Let's evaluate $f(-x)$ for $f(x) = x^3 - x$. $f(-x) = (-x)^3 - (-x) = -x^3 + x$. Now let's compare $f(-x)$ with $-f(x)$. $-f(x) = -(x^3 - x) = -x^3 + x$. Since $f(-x) = -f(x)$, the function $f(x) = x^3 - x$ is an odd function.

4. Let $f(x) = x^2$. Which of the following is not a property of $f(x)$ on $\mathbb{R}$?

   1. It is not injective.
   2. It is not surjective.
   3. It is neither even nor odd.
   4. Its range is $[0, \infty)$.
   Show answer

   Answer: C. It is neither even nor odd.

   Let $f(x) = x^2$ on the domain $\mathbb{R}$. 1. Injectivity: $f(2) = 4$ and $f(-2) = 4$. Since $2 eq -2$, $f$ is not injective. This statement is true. 2. Surjectivity: The range of $f(x) = x^2$ on $\mathbb{R}$ is $[0, \infty)$. The codomain is assumed to be $\mathbb{R}$. Since the range is not equal to the codomain (e.g., -1 is in the codomain but not in the range), $f$ is not surjective onto $\mathbb{R}$. This statement is true. 3. Even/Odd: $f(-x) = (-x)^2 = x^2 = f(x)$. So, $f$ is an even function. This statement claims $f$ is neither even nor odd, which is false. Therefore, this is the statement that is NOT a property of $f(x)$. 4. Range: As established, the range of $f(x) = x^2$ on $\mathbb{R}$ is indeed $[0, \infty)$. This statement is true. The question asks which of the following is NOT a property. The false statement is that $f$ is neither even nor odd.

5. Let $f(x) = x^3$. Which of the following statements is false?

   1. $f$ is injective on $\mathbb{R}$.
   2. $f$ is surjective on $\mathbb{R}$.
   3. $f$ is an odd function.
   4. $f$ is a constant function.
   Show answer

   Answer: D. $f$ is a constant function.

   Let $f(x) = x^3$. We analyze each statement for $f: \mathbb{R} o \mathbb{R}$. 1. Injectivity: If $f(x_1) = f(x_2)$, then $x_1^3 = x_2^3$, which implies $x_1 = x_2$. So, $f$ is injective. This statement is true. 2. Surjectivity: For any $y \in \mathbb{R}$, there exists $x = \sqrt[3]{y} \in \mathbb{R}$ such that $f(x) = x^3 = (\sqrt[3]{y})^3 = y$. So, $f$ is surjective. This statement is true. 3. Odd function: $f(-x) = (-x)^3 = -x^3 = -f(x)$. So, $f$ is an odd function. This statement is true. 4. Constant function: A constant function has the same output for every input. For $f(x) = x^3$, $f(1) = 1$ and $f(2) = 8$. Since $f(1) eq f(2)$, $f$ is not a constant function. This statement is false. The question asks for the false statement.

More questions in this quiz

Answers and explanations are in the full quiz.

1. If $A = \{1, 2, 3\}$ and $B = \{a, b, c\}$, the number of distinct relations from $A$ to $B$ is:
2. Let $R$ be the relation defined on the set of integers $\mathbb{Z}$ by $aRb$ if and only if $a-b$ is an even integer. Which of the following is true about $R$?
3. Consider the function $f(x) = \sqrt{1 - x^2}$. What is the domain of $f(x)$?
4. Let $f(x) = x^2 + 1$. Which of the following is the inverse function $f^{-1}(x)$ if the domain of $f$ is restricted to $[0, \infty)$?
5. If $f(x) = \frac{x+1}{x-1}$ and $g(x) = \frac{x-1}{x+1}$, then $(f \circ g)(x)$ is:
6. Let $f: \mathbb{N} o \mathbb{N}$ be defined by $f(n) = n+1$. What is the range of $f$?
7. Which of the following functions is an odd function?
8. If $f(x) = \log_a x$, where $a > 0$ and $a eq 1$, then $f(xy)$ is equal to:
9. Consider the function $f(x) = |x|$. For which of the following sets is $f$ injective?
10. Let $f(x) = x^3$. Is $f$ an injective function?
11. If $f(x) = x^2+1$ and $g(x) = 3x-2$, find $(f \circ g)(x)$.
12. Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$. The number of one-to-one functions from $A$ to $B$ is:
13. If $f(x) = \frac{1}{x-2}$ and $g(x) = \frac{x+1}{x-1}$, find the domain of $f \circ g$.
14. Let $f(x) = x^2$. Find the domain of $f^{-1}(x)$ if the domain of $f$ is restricted to $[0, \infty)$.
15. If $f(x) = \frac{1}{x}$, then $f(f(f(x)))$ is:
16. Let $f(x) = \frac{x}{x+1}$. What is $f(f(x))$?
17. Let $f(x) = x^2 - 2x + 1$. Find the value of $x$ for which $f(x) = 0$.
18. Which of the following relations is an equivalence relation on the set of integers $\mathbb{Z}$?
19. Let $f(x) = \log_2 x$. What is $f(x) + f(x^2)$ in terms of $f(x)$?
20. Let $f(x) = \sqrt{x^2}$. What is the domain of $f$?
21. Let $f: A o B$ be a function. $f$ is surjective if:
22. If $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x}$, what is $(f \cdot g)(x)$?
23. Let $f(x) = x+1$ and $g(x) = 2x$. Find $(g \circ f)(x)$.
24. Consider the relation $R = \{(a, b) \mid a, b \in \mathbb{Z} ext{ and } a^2 - b^2 ext{ is even}\}$. Is $R$ reflexive?
25. Let $f(x) = \sin(x)$. Which of the following is true?
26. Let $f(x) = x^2$ and $g(x) = x+1$. Find $(f-g)(x)$.
27. Let $f(x) = 3x$. Is $f$ injective?
28. Let $f(x) = e^x$. Which of the following is true?
29. Let $A = \{1, 2\}$, $B = \{3, 4\}$. The number of possible relations from $A$ to $B$ is:
30. If $f(x) = x^3$, find $f(f(x))$.
31. Let $f(x) = \frac{1}{x+1}$. What is $f(f(f(x)))$?
32. Let $f(x) = 2x+1$ and $g(x) = \frac{x-1}{2}$. What is $(g \circ f)(x)$?
33. If $f(x) = \log_e x$, what is $f(x) + f(y)$ equal to?
34. Let $f(x) = 2^x$. Then $f(x+y)$ is equal to:
35. If $f(x) = x^2 + 1$ and $g(x) = x^2 - 1$, then $(f \circ g)(x)$ is:
36. Which of the following is a one-to-one correspondence from $\mathbb{R}$ to $\mathbb{R}$?
37. Let $f(x) = x^2 - 4$. Find the domain of $f^{-1}(x)$ if the domain of $f$ is restricted to $[2, \infty)$.
38. Let $f(x) = \frac{1}{x}$. Which of the following is the domain of $f(f(x))$?
39. Let $f(x) = x^3$. Then $f$ is:
40. If $f(x) = x^2$, and the domain of $f$ is restricted to $[-1, 1]$, what is the range of $f$?
41. Let $f(x) = \frac{1}{x-a}$ and $g(x) = \frac{x}{x-1}$. If $(f \circ g)(x)$ is defined for all $x$ except $x=1$ and $x=3$, what is the value of $a$?
42. Let $A = \{1, 2, 3\}$, $B = \{a, b\}$. The number of functions from $A$ to $B$ is:
43. If $f(x) = \sqrt{x}$ and $g(x) = x^2$, what is the domain of $(f \circ g)(x)$?
44. Which function is NOT a rational function?
45. Let $f(x) = x^2$. Find the inverse relation of $f(x)$.
46. Let $f(x) = x^3 - 1$. Find the value of $x$ for which $f(x) = 7$.
47. Let $f(x) = \frac{1}{x}$. Then $f(x+h) - f(x)$ is:
48. Consider the relation $R = \{(a, b) \mid a, b \in \mathbb{Z} ext{ and } a - b ext{ is divisible by } 3\}$. Which of the following is true?
49. Let $f(x) = \frac{x}{x+1}$. Find $f^{(n)}(x)$, the $n$-th iteration of $f$, for small values of $n$, and identify the pattern.
50. Let $f(x) = x^2$ and $g(x) = 2x$. What is the domain of $(f/g)(x)$?
51. Let $f(x) = \sin x$. Find $f(x + \pi)$.
52. Let $f(x) = \sqrt{x-2}$ and $g(x) = x^2$. What is the domain of $(f \circ g)(x)$?

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