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Relations and Functions Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Relations and Functions Set-5

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

The chapter “Relations and Functions” builds the foundation for understanding how mappings behave (injective, surjective, bijective), how functions can be composed, and how algebraic and set-based conditions translate into function properties. These ideas are heavily used in board exams and are also frequent in competitive exams through questions on inverse/involution functions, functional equations, continuity/functional forms, and counting functions with constraints.

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Q1. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=ax+b$ where $a,b\in\mathbb{R}$ . For which pairs $(a,b)$ does $f(f(x))=x$ for all real $x$ ?

(A)

$a=1$ with any real $b$ .

(B)

$a^2=1$ and $b=0$ .

(C)

$a=1,\;b=0$ or $a=-1$ with $b\in\mathbb{R}$ .

(D)

$a=-1,\;b=0$ only.

Q2. For which real values of $p$ is the function $f(x)=x^3+px$ injective on $\mathbb{R}$ ?

(A)

$p\ge 0$ .

(B)

$p>0$ .

(C)

$p\le 0$ .

(D)

$p\in\mathbb{R}$ (all real $p$ ).

Q3. Let $f,g:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=ax+b$ and $g(x)=cx+d$ , where $a,b,c,d\in\mathbb{R}$ . Find the necessary and sufficient condition on $a,b,c,d$ for $f\circ g=g\circ f$ for all real $x$ .

(A)

$a=c$ .

(B)

$ad-bc=0$ .

(C)

$(a-1)(c-1)=0$ .

(D)

$d(a-1)=b(c-1)$ .

Q4. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous and satisfy $f(f(x))=x$ for all real $x$ . Which of the following statements must hold?
(i) $f$ is bijective.
(ii) $f$ is either strictly increasing or strictly decreasing.
(iii) $f(x)=x$ for all $x$ .

(A)

Only (i) is true.

(B)

(i) and (ii) only.

(C)

(i) and (iii) only.

(D)

(i), (ii) and (iii) are all true.

Q5. Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(f(x))=x$ for all real $x$ , and suppose $f(x)\ge x$ for every real $x$ . Then which of the following is true?

(A)

$f(x)=x$ for all real $x$ .

(B)

$f(x)>x$ for all real $x$ .

(C)

$f$ is strictly increasing but not the identity.

(D)

No such function exists.

Q6. Let $f:\mathbb{R}\to\mathbb{R}$ be defined by

f(x)=x^2-4x+5.

Find the set $f^{-1}([5,8])$ .

(A)

$[\,2-\sqrt{7},\,0)\cup(4,\,2+\sqrt{7}]$

(B)

$[\,2-\sqrt{7},\,0\,]\cup[\,4,\,2+\sqrt{7}\,]$

(C)

$(-\infty,\,2-\sqrt{7}]\cup[0,\,2+\sqrt{7}]$

(D)

$[0,\,4]$

Q7. Let $f:\mathbb{R}\to\mathbb{R}$ be linear of the form $f(x)=mx+n$ and satisfy

f(f(x))=4x+3\quad\text{for all }x\in\mathbb{R}.

Which of the following describes all possible linear functions $f$ ?

(A)

Both $f(x)=2x+1$ and $f(x)=-2x-3$

(B)

$f(x)=2x+1$ only

(C)

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

(D)

No linear function exists

Q8. For a fixed real $t$ with $0<t<\tfrac{1}{2}$ , consider the equation

\frac{x}{1+x^2}=t.

How many distinct real solutions does this equation have?

(A)

No real solution

(B)

Exactly one real solution

(C)

Infinitely many real solutions

(D)

Exactly two real solutions

Q9. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous and satisfy for all real $x,y$ the identity

f(x+y)+f(x-y)=2\bigl(f(x)+f(y)\bigr),\qquad f(0)=0.

Then $f$ must be of which form?

(A)

$f(x)\equiv 0$ only

(B)

$f(x)=ax^3+bx$ for some constants $a,b\in\mathbb{R}$

(C)

$f(x)=ax^2\,$ for some constant $a\in\mathbb{R}$

(D)

$f(x)=ax\,$ for some constant $a\in\mathbb{R}$

Q10. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous and satisfy

f(x+y)=f(x)+f(y)+2xy\quad\text{for all }x,y\in\mathbb{R},

with $f(1)=1$ . Then $f(x)=$

(A)

$x^2+x$

(B)

$x^2$

(C)

$x^2+2x$

(D)

$0$

Q11. Let $f:[0,\infty)\to\mathbb{R}$ be defined by $f(x)=\sqrt{x}$ and $g:\mathbb{R}\setminus\{-1\}\to\mathbb{R}$ by $g(x)=\dfrac{2x-5}{x+1}$ . The domain of the composite function $f\circ g$ is

(A)

$(-\infty,-1]\cup\left(\frac{5}{2},\infty\right)$

(B)

$(-\infty,-1)\cup\left[\frac{5}{2},\infty\right)$

(C)

$\left[\frac{5}{2},\infty\right)$

(D)

$(-\infty,-1)\cup\left(\frac{5}{2},\infty\right)$

Q12. For real parameter $a$ , define $f_a:\mathbb{R}\setminus\{a\}\to\mathbb{R}$ by $f_a(x)=\dfrac{ax+1}{x-a}$ . For which real $a$ does $f_a\circ f_a$ equal the identity on its domain?

(A)

$a=0$ only

(B)

$a=1$ or $a=-1$

(C)

No real $a$ makes $f_a$ involutive

(D)

All real $a$

Q13. Let $A=\{1,2,3,4,5,6\}$ . A function $f:A\to A$ satisfies $f(f(x))=x$ for all $x\in A$ . Let $k$ be the number of elements fixed by $f$ (i.e., $f(x)=x$ ). Which of the following is the set of all possible values of $k$ ?

(A)

$\{0,2,4,6\}$

(B)

$\{1,3,5\}$

(C)

$\{0,1,2,3,4,5,6\}$

(D)

$\{2,4\}$

Q14. Define $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=1-x$ if $x\in\mathbb{Q}$ and $f(x)=x$ if $x\notin\mathbb{Q}$ . Which of the following statements are true?
(I) $f$ is bijective.
(II) $f\circ f$ is the identity on $\mathbb{R}$ .
(III) $f$ is continuous only at $x=\dfrac{1}{2}$ .

(A)

Only (I) and (II) are true

(B)

Only (I) is true

(C)

(I), (II) and (III) are all true

(D)

(II) and (III) are true but (I) is false

Q15. How many functions $f:\{1,2,3,4\}\to\{1,2,3,4\}$ satisfy $f(f(x))=x$ for all $x$ ?

(A)

$6$

(B)

$10$

(C)

$24$

(D)

$16$

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