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Relations and Functions Set-3

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

This chapter is central to understanding how inputs map to outputs and how different types of functions (like injective, surjective, bijective) behave. For board exams, it builds the foundation for solving inverse-function, composition, and identity/idempotent relations. For competitive exams (JEE/NEET/CBSE), these same ideas are frequently used to test logical reasoning quickly through algebraic conditions, equivalence relations, and function composition properties.

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Q1. Let $f:\mathbb{R}\setminus\{2\}\to\mathbb{R}$ be defined by $f(x)=\dfrac{3x+4}{x-2}$ . Then $f^{-1}(5)=$

(A)

$3$

(B)

$-1$

(C)

$7$

(D)

$\dfrac{14}{3}$

Q2. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by

\begin{aligned} f(x)=x^2+1 & \quad\text{if } x\ge 0\\ f(x)=ax+b & \quad\text{if } x<0 \end{aligned}

For which real pairs $(a,b)$ is $f$ one-to-one?

(A)

$\{(a,b)\in\mathbb{R}^2:\;a>0,\;b\le 1\}$

(B)

$\{(a,b)\in\mathbb{R}^2:\;a\ge 0,\;b<1\}$

(C)

$\{(a,b)\in\mathbb{R}^2:\;a>0,\;b<1\}$

(D)

$\{(a,b)\in\mathbb{R}^2:\;a>0,\;b\ge 1\}$

Q3. Let $A=\{1,2,3\}$ and $B=\{a,b,c,d\}$ . Suppose $f:A\to B$ is an injective function. How many functions $g:B\to A$ satisfy $g\circ f=\mathrm{id}_A$ ?

(A)

$1$

(B)

$4$

(C)

$9$

(D)

$3$

Q4. Let $f:\mathbb{R}\to(-1,1)$ be defined by $f(x)=\dfrac{x}{1+|x|}$ . The inverse $f^{-1}:(-1,1)\to\mathbb{R}$ is given by:

(A)

\begin{aligned} f^{-1}(y)=\frac{y}{1+y} & \quad\text{if } y\ge 0\\ f^{-1}(y)=\frac{y}{1-y} & \quad\text{if } y<0 \end{aligned}

(B)

\begin{aligned} f^{-1}(y)=\frac{y}{1-y} & \quad\text{if } y\ge 0\\ f^{-1}(y)=\frac{y}{1+y} & \quad\text{if } y<0 \end{aligned}

(C)

\begin{aligned} f^{-1}(y)=\frac{y}{1+|y|} & \quad\text{for all } y\in(-1,1) \end{aligned}

(D)

\begin{aligned} f^{-1}(y)=\frac{y}{1+y} & \quad\text{if } y>0\\ f^{-1}(y)=\frac{y}{1-y} & \quad\text{if } y\le 0 \end{aligned}

Q5. Assertion: If $f:A\to B$ and $g:B\to C$ are functions and $g\circ f$ is bijective, then $f$ is injective and $g$ is surjective. Reason: Because $(g\circ f)$ being injective forces $f$ to be injective, and $(g\circ f)$ being surjective forces $g$ to be surjective.

(A)

Both Assertion and Reason are true and Reason is the correct explanation of the Assertion.

(B)

Both Assertion and Reason are true but Reason is not the correct explanation of the Assertion.

(C)

Assertion is true but Reason is false.

(D)

Both Assertion and Reason are false.

Q6. Let $f:\mathbb{R}\to(-1,1)$ be defined by $f(x)=\dfrac{x}{1+|x|}$ . Which of the following is true about $f$ ?

(A)

$f$ is injective but not surjective.

(B)

$f$ is surjective but not injective.

(C)

$f$ is bijective (both one-one and onto).

(D)

$f$ is neither injective nor surjective.

Q7. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by the piecewise rule

\begin{aligned} f(x)=x^2 & \text{ if } x\ge 0\\ f(x)=2x+1 & \text{ if } x<0 \end{aligned}

Which of the following properties does $f$ have?

(A)

$f$ is surjective but not injective.

(B)

$f$ is injective but not surjective.

(C)

$f$ is bijective.

(D)

$f$ is neither injective nor surjective.

Q8. Let $f,g:\mathbb{R}\to\mathbb{R}$ be linear functions $f(x)=ax+b$ and $g(x)=cx+d$ with $a\neq1,\;c\neq1$ . If $f\circ g=g\circ f$ for all real $x$ , which of the following must hold?

(A)

$b=d$

(B)

$a=c$

(C)

$\dfrac{b}{a-1}=-\dfrac{d}{c-1}$

(D)

$\dfrac{b}{a-1}=\dfrac{d}{c-1}$

Q9. Assertion (A): The relation $R$ on $\mathbb{R}$ defined by $xRy\iff x-y\in\mathbb{Q}$ is an equivalence relation.
Reason (R): Every equivalence class of $R$ is countable. Which one is correct?

(A)

A is true, R is false.

(B)

A is true, R is true but R does not explain A.

(C)

A is true, R is true and R explains A.

(D)

A is false, R is true.

Q10. Let $f(x)$ be a polynomial with real coefficients satisfying $f(f(x))=f(x)$ for all real $x$ . Which of the following describes all such polynomials?

(A)

All constant polynomials and the identity polynomial $f(x)=x$ .

(B)

All constant polynomials and all linear polynomials $f(x)=ax+b$ with $a=1$ .

(C)

All polynomials whose degree is $0$ or a power of $2$ .

(D)

Only constant polynomials.

Q11. Let $f:\mathbb{R}\to\mathbb{R}$ be a linear function $f(x)=ax+b$ such that

f(f(x))=4x+3

for all real $x$ and $f(0)>0$ . Then $f(x)$ equals:

(A)

$f(x)=-2x-3$

(B)

$f(x)=2x-1$

(C)

$f(x)=2x+1$

(D)

$f(x)=-2x+3$

Q12. Let $R$ be a relation on $\mathbb{Z}$ defined by $aRb$ iff $a^2\equiv b^2\pmod{5}$ . The number of distinct equivalence classes of $R$ is:

(A)

$3$

(B)

$5$

(C)

$4$

(D)

$Infinite

Q13. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by

f(x)=\frac{x^2-x}{x^2+1}.

The range of $f$ is:

(A)

$(-1,1)$

(B)

$\displaystyle\left(\frac{1-\sqrt{2}}{2},\frac{1+\sqrt{2}}{2}\right)$

(C)

$\displaystyle\left[-\tfrac{1}{2},\tfrac{3}{2}\right]$

(D)

$\displaystyle\left[\frac{1-\sqrt{2}}{2},\frac{1+\sqrt{2}}{2}\right]$

Q14. Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(f(x))=f(x)$ for all real $x$ . Which one of the following statements is necessarily true?

(A)

$f$ must be the identity function $f(x)=x$ for all $x$

(B)

For every $y\in\operatorname{Im}(f)$ , $f(y)=y$ , i.e., $f$ acts as the identity on its image

(C)

$f$ must be a constant function

(D)

$f$ is necessarily injective

Q15. Define a relation $R$ on $\mathbb{R}$ by $xRy$ iff $xy\ge 0$ . Which statement is correct?

(A)

$R$ is an equivalence relation on $\mathbb{R}\setminus\{0\}$ but not an equivalence relation on $\mathbb{R}$

(B)

$R$ is an equivalence relation on $\mathbb{R}$

(C)

$R$ is not an equivalence relation on either $\mathbb{R}$ or $\mathbb{R}\setminus\{0\}$

(D)

$R$ is a partial order on $\mathbb{R}$

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