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Relations and Functions Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Relations and Functions Set-6

Practice Important MCQs for Relations and Functions — Mathematics, Class 12.

The chapter “Relations and Functions” is crucial for both board exams and competitive tests because it builds the foundation for understanding how mappings behave under composition, inverses, injectivity/surjectivity, and equivalence classes. Concepts like inverse functions, functional equations, and classification of relations directly appear in higher-difficulty problems across CBSE and entrance exams.

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Q1. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^3+x$ . Which of the following is true?

(A)

$f$ is not injective.

(B)

$f$ is bijective and $f^{-1}(0)=0$ .

(C)

$f$ is injective but not surjective.

(D)

$f$ is surjective but not injective.

Q2. Let $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ be defined by $f(x)=x+\dfrac{1}{x}$ . The range of $f$ is

(A)

$(-\infty,-2)\cup[2,\infty)$ .

(B)

$(-\infty,-2]\cup(2,\infty)$ .

(C)

$(-\infty,-2)\cup(2,\infty)$ .

(D)

$(-\infty,-2]\cup[2,\infty)$ .

Q3. Let $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=x^2$ and let $A=[-1,2]$ . Then

(A)

$f\big(f^{-1}(A)\big)=[0,2]\quad\text{and}\quad f^{-1}\big(f(A)\big)=[-2,2]$ .

(B)

$f\big(f^{-1}(A)\big)=A\quad\text{and}\quad f^{-1}\big(f(A)\big)=A$ .

(C)

$f\big(f^{-1}(A)\big)=[-1,2]\quad\text{and}\quad f^{-1}\big(f(A)\big)=[-2,2]$ .

(D)

$f\big(f^{-1}(A)\big)=[0,2]\quad\text{and}\quad f^{-1}\big(f(A)\big)=[-1,2]$ .

Q4. Let $A,B,C$ be non-empty sets and $f:A\to B$ , $g:B\to C$ be functions. Consider the statements:
P: "If $g\circ f$ is bijective then $f$ is bijective."
Q: "If $g\circ f$ is bijective then $g$ is bijective." Which option correctly describes P and Q?

(A)

Both P and Q are true.

(B)

P is true but Q is false.

(C)

Both P and Q are false.

(D)

P is false but Q is true.

Q5. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^3-3ax+1$ , where $a\in\mathbb{R}$ . For which real values of $a$ is $f$ one-to-one?

(A)

$(-\infty,0)$ .

(B)

$(-\infty,0]$ .

(C)

$[0,\infty)$ .

(D)

$(-\infty,1]$ .

Q6. Let $f:\mathbb{R}\setminus\{2\}\to\mathbb{R}\setminus\{1\}$ be defined by $f(x)=\dfrac{x+1}{x-2}$ . Find $f^{-1}(x)$ .

(A)

$\dfrac{x-1}{2x+1}$

(B)

$\dfrac{2x+1}{x-1}$

(C)

$\dfrac{x+2}{x-1}$

(D)

$\dfrac{1-2x}{x-1}$

Q7. Let $f:\mathbb{R}\setminus\{1\}\to\mathbb{R}$ be given by $f(x)=\dfrac{x}{1-x}$ . For $n\in\mathbb{N}$ let $f^{n}$ denote the $n$ -fold composition of $f$ with itself. Then, for $x\neq \dfrac{1}{k}$ ( $k=1,2,\dots,n$ ), $f^{n}(x)=$

(A)

$\dfrac{x}{1+(n-1)x}$

(B)

$\dfrac{nx}{1-x}$

(C)

$\dfrac{x}{1-x^{n}}$

(D)

$\dfrac{x}{1-nx}$

Q8. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^{3}+3x$ . Which of the following is true?

(A)

$f$ is bijective on $\mathbb{R}$

(B)

$f$ is injective but not surjective

(C)

$f$ is surjective but not injective

(D)

$f$ is neither injective nor surjective

Q9. Assertion (A): For functions $f:A\to B$ and $g:B\to C$ , if $g\circ f$ is bijective then $f$ is bijective.
Reason (R): If $g\circ f$ is bijective then $f$ has a left inverse (so $f$ is injective) and $g$ has a right inverse (so $g$ is surjective).

(A)

Both A and R are true and R is the correct explanation of A.

(B)

A is true but R is false.

(C)

A is false but R is true.

(D)

Both A and R are false.

Q10. Let $f:\mathbb{R}\to(-1,1)$ be defined by $f(x)=\dfrac{x}{1+|x|}$ . Then the inverse function $f^{-1}:(-1,1)\to\mathbb{R}$ is

(A)

$\dfrac{x}{1+x}$

(B)

$\dfrac{x}{1-|x|}$

(C)

$\dfrac{x}{1-x}$

(D)

$\dfrac{x}{1+|x|}$

Q11. Let $A=\{1,2,3,4\}$ . A function $f:A\to A$ satisfies $f(f(x))=x$ for all $x\in A$ . How many such functions $f$ exist?

(A)

$6$

(B)

$9$

(C)

$10$

(D)

$12$

Q12. Let $f:\mathbb{R}\setminus\{1\}\to\mathbb{R}\setminus\{2\}$ be defined by $f(x)=\dfrac{2x+3}{x-1}$ . Then $f^{-1}(x)$ equals:

(A)

$\dfrac{3-2x}{x-1}$

(B)

$\dfrac{x+3}{x-2}$

(C)

$\dfrac{2x-3}{x+1}$

(D)

$\dfrac{x-3}{2-x}$

Q13. Define a relation $R$ on $\mathbb{Z}$ by $xRy\iff 7\mid (x^2-y^2)$ . How many distinct equivalence classes does $R$ have?

(A)

$3$

(B)

$5$

(C)

$7$

(D)

$4$

Q14. Let $f:A\to B$ and $g:B\to C$ be functions.
Statement (A): If both $f$ and $g$ are bijections then $g\circ f$ is a bijection.
Statement (R): If $g\circ f$ is a bijection then both $f$ and $g$ must be bijections.

(A)

A is true and R is false.

(B)

Both A and R are true, but R is not a correct explanation of A.

(C)

Both A and R are true and R is a correct explanation of A.

(D)

A is false and R is true.

Q15. Let $A$ be a set with $|A|=4$ . A function $f:A\to A$ is idempotent if $f(f(x))=f(x)$ for all $x\in A$ . How many idempotent functions exist from $A$ to $A$ ?

(A)

$16$

(B)

$24$

(C)

$41$

(D)

$64$

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