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Relations and Functions Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Relations and Functions Set-4

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

This chapter is crucial because it builds the foundation of function behavior (domain, range, injectivity, surjectivity, inverse functions) and the structure of relations. These concepts repeatedly appear in board exams and competitive tests, especially in questions involving inverse functions, mappings, and function properties like injective/surjective/bijective and idempotent/involution functions.

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Q1. Let $f:\mathbb{R}\setminus\{1\}\to\mathbb{R}\setminus\{2\}$ be defined by $f(x)=\dfrac{2x+3}{x-1}$ . Find $f^{-1}(x)$ and its domain.

(A)

$f^{-1}(x)=\dfrac{x-3}{x+2},\; x\ne-2$

(B)

$f^{-1}(x)=\dfrac{x+2}{x-3},\; x\ne3$

(C)

$f^{-1}(x)=\dfrac{x+3}{x-2},\; x\ne2$

(D)

$f^{-1}(x)=\dfrac{2x-3}{x+1},\; x\ne-1$

Q2. Determine the range of the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=\dfrac{x}{1+x^2}$ .

(A)

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

(B)

$\displaystyle \left(-\tfrac{1}{2},\tfrac{1}{2}\right)$

(C)

$\displaystyle [0,1]$

(D)

$\displaystyle \left(-\infty,\tfrac{1}{2}\right]$

Q3. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x+\sin x$ . Which of the following statements is true?

(A)

$f$ is injective but not surjective

(B)

$f$ is surjective but not injective

(C)

$f$ is neither injective nor surjective

(D)

$f$ is bijective

Q4. Find the range of the function $g:\mathbb{R}\to\mathbb{R}$ defined by $g(x)=\dfrac{x^2+1}{x^2+2x+2}$ .

(A)

$\displaystyle \left(-\infty,\;\dfrac{3+\sqrt{5}}{2}\right]$

(B)

$\displaystyle \left[\dfrac{3-\sqrt{5}}{2},\;\dfrac{3+\sqrt{5}}{2}\right]$

(C)

$\displaystyle [0,1]$

(D)

$\displaystyle \left(\dfrac{3-\sqrt{5}}{2},\;\dfrac{3+\sqrt{5}}{2}\right)$

Q5. Let $A$ be a finite set with $|A|=n\ge1$ . How many functions $f:A\to A$ satisfy $f\circ f=f$ ?

(A)

$n^n$

(B)

$\displaystyle \sum_{k=1}^{n}\binom{n}{k}(k-1)^{\,n-k}$

(C)

$\displaystyle \sum_{k=1}^{n}\binom{n}{k}k^{\,n-k}$

(D)

$\displaystyle \sum_{k=0}^{n}\binom{n}{k}k!$

Q6. Let $A=\{1,2,3\}$ and $B=\{a,b,c,d,e\}$ . How many one-one functions from $A$ to $B$ are there?

(A)

$125$

(B)

$10$

(C)

$60$

(D)

$20$

Q7. Let $S=\{1,2,3,4\}$ . How many functions $f:S\to S$ satisfy $f(f(x))=x$ for all $x\in S$ ?

(A)

$10$

(B)

$9$

(C)

$12$

(D)

$15$

Q8. 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)

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

(B)

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

(C)

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

(D)

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

Q9. Let $S$ be a set with $4$ elements. How many functions $f:S\to S$ satisfy $f(f(x))=f(x)$ for all $x\in S$ ?

(A)

$64$

(B)

$41$

(C)

$24$

(D)

$16$

Q10. Assertion (A): Let $f:A\to B$ be a function. If there exist $g:B\to A$ and $h:B\to A$ such that $g\circ f=\mathrm{id}_A$ and $f\circ h=\mathrm{id}_B$ , then $g=h$ and $f$ is bijective.
Reason (R): A left inverse of $f$ implies $f$ is injective, a right inverse implies $f$ is surjective; existence of both implies $f^{-1}$ exists and equals both inverses.

(A)

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

(B)

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

(C)

A is true but R is false.

(D)

A is false but R is true.

Q11. Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3+x$ . Which of the following is true?

(A)

$f$ is one-to-one but not onto.

(B)

$f$ is both one-to-one and onto.

(C)

$f$ is onto but not one-to-one.

(D)

$f$ is neither one-to-one nor onto.

Q12. Find the range of the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x^2+2x+3}{x^2+1}$ .

(A)

$(1,3)$

(B)

$[1,3]$

(C)

$(-\infty,\infty)$

(D)

$\displaystyle\big[\,2-\sqrt{2}\,,\,2+\sqrt{2}\,\big]$

Q13. Let $A$ be a set with $|A|=5$ . How many functions $f:A\to A$ satisfy $f(f(x))=f(x)$ for all $x\in A$ ?

(A)

$196$

(B)

$150$

(C)

$125$

(D)

$312$

Q14. Assertion (A): The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x+\sin x$ is bijective.
Reason (R): $f'(x)=1+\cos x\ge 0$ for all $x$ , with $f'(x)=0$ only at isolated points, so for $a<b$ we have $f(b)-f(a)=\displaystyle\int_a^b f'(x)\,dx>0$ , and $\lim_{x\to\infty}f(x)=\infty,\ \lim_{x\to-\infty}f(x)=-\infty$ .

(A)

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

(B)

Both A and R are true but R is NOT the correct explanation of A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q15. Let $f:B\to A$ and $g:A\to B$ be functions (non-empty sets) such that $f\circ g=\mathrm{id}_A$ . Which statement must be true?

(A)

$g$ is surjective and $f$ need not be injective.

(B)

$g$ is injective and $f$ is surjective.

(C)

$g$ is injective and $f$ is injective.

(D)

Neither $g$ must be injective nor $f$ must be surjective.

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