Class 11 Mathematics Functions MCQs Set 02

Question. The domain of \( f(x) = \frac{1}{\sqrt{|x| - x}} \) is
(a) \( (-\infty, 0) \)
(b) \( (0, \infty) \)
(c) \( (1, \infty) \)
(d) \( (-\infty, \infty) \)
Answer: (a) \( (-\infty, 0) \)

Question. The domain of \( f(x) = \frac{1}{[x] - x} \) is
(a) R
(b) Z
(c) R - Z
(d) Q-{0}
Answer: (c) R - Z

Question. The domain of \( f(x) = \sqrt{x - 2} + \frac{1}{\log(4 - x)} \) is
(a) \( [2, \infty) \)
(b) \( (-\infty, 4) \)
(c) \( [2, 3) \cup (3, 4) \)
(d) \( [3, \infty) \)
Answer: (c) \( [2, 3) \cup (3, 4) \)

Question. The domain of \( f(x) = e^{\sqrt{x}} + \cos x \) is
(a) \( (-\infty, \infty) \)
(b) \( [0, \infty) \)
(c) (0,1)
(d) \( (1, \infty) \)
Answer: (b) \( [0, \infty) \)

Question. The domain of \( \log_a \sin^{-1} x \) is (a>0, a \( \neq \) 1)
(a) \( 0 < x \le 1 \)
(b) \( 0 \le x \le 1 \)
(c) \( 0 \le x < 1 \)
(d) \( 0 < x < 1 \)
Answer: (a) \( 0 < x \le 1 \)

Question. The domain of \( \cosh^{-1} 5x \) is
(a) R
(b) \( [0, \infty) \)
(c) \( \left(\frac{1}{5}, \infty\right) \)
(d) \( \left[\frac{1}{5}, \infty\right) \)
Answer: (d) \( \left[\frac{1}{5}, \infty\right) \)

Question. For which Domain, the functions \( f(x) = 2x^2 - 1 \) and \( g(x) = 1 - 3x \) are equal to
(a) R
(b) \( \left\{\frac{1}{2}, -2\right\} \)
(c) \( \left\{\frac{1}{2}, 2\right\} \)
(d) \( \left[\frac{1}{2}, 2\right] \)
Answer: (b) \( \left\{\frac{1}{2}, -2\right\} \)

Range of the function :

Question. The domain and range of the real function f defined by \( f(x) = \frac{4 - x}{x - 4} \) is given by
(a) Domain = R, Range = {–1, 1}
(b) Domain = R – {1}, Range = R
(c) Domain = R – {4}, Range = {–1}
(d) Domain = R – {–4}, Range = {–1, 1}
Answer: (c) Domain = R – {4}, Range = {–1}

Question. Range of \( f(x) = \frac{1}{1 - 2\cos x} \) is
(a) \( \left[\frac{1}{3}, 1\right] \)
(b) \( \left[-1, \frac{1}{3}\right] \)
(c) \( (-\infty, -1] \cup \left[\frac{1}{3}, \infty\right) \)
(d) \( \left[-\frac{1}{3}, 1\right] \)
Answer: (c) \( (-\infty, -1] \cup \left[\frac{1}{3}, \infty\right) \)

Question. The range of \( f(x) = x^2 + x + 1 \) is
(a) \( \left[\frac{3}{4}, \infty\right) \)
(b) \( [0, \infty) \)
(c) \( [1, \infty) \)
(d) \( \left[\frac{1}{4}, \infty\right) \)
Answer: (a) \( \left[\frac{3}{4}, \infty\right) \)

Question. The domain and range of the function f given by \( f(x) = 2 - |x - 5| \) is
(a) Domain = \( R^+ \), Range = \( (-\infty, 1] \)
(b) Domain = R, Range = \( (-\infty, 2] \)
(c) Domain = R, Range = \( (-\infty, 2) \)
(d) Domain = \( R^+ \), Range = \( (-\infty, 2] \)
Answer: (b) Domain = R, Range = \( (-\infty, 2] \)

Question. \( f = \left\{\left(x, \frac{x^2}{x^2 + 1}\right) : x \in R\right\} \), be a function R into R, range of 'f'
(a) [0,1)
(b) \( (-\infty, \infty) \)
(c) \( (0, \infty) \)
(d) \( R^+ \)
Answer: (a) [0,1)

Question. Range of the function \( f(x) = \sqrt{[x] - x} \) is
(a) R
(b) {1}
(c) {0}
(d) \( [0, \infty) \)
Answer: (c) {0}

Question. Let A = {9, 10, 11, 12, 13} and \( f : A \rightarrow N \) be defined by f(n) = highest prime factor of n, then its range is
(a) {13}
(b) {3, 5, 11, 13}
(c) {11, 13}
(d) {2, 3, 5, 11}
Answer: (b) {3, 5, 11, 13}

Question. The range of \( f(x) = \frac{x^2 - x + 1}{x^2 + x + 1} \) is
(a) \( \left[\frac{1}{3}, 3\right] \)
(b) \( \left[\frac{1}{2}, 2\right] \)
(c) [0,1]
(d) [-1,1]
Answer: (a) \( \left[\frac{1}{3}, 3\right] \)

Types of functions :

Question. The equivalent function of \( \log x^2 \) is
(a) \( 2\log x \)
(b) \( 2\log|x| \)
(c) \( |\log x^2| \)
(d) \( (\log x)^2 \)
Answer: (b) \( 2\log|x| \)

Question. The number of linear functions which map \( [-1,1] \) to \( [0,2] \) are
(a) One
(b) Two
(c) Four
(d) Three
Answer: (b) Two

Question. If A = (3,81) and \( f : A \rightarrow B \) is a surjection defined by \( f(x) = \log_3 x \) then B =
(a) [1, 4]
(b) (1, 4]
(c) (1, 4)
(d) [1, \( \infty \))
Answer: (c) (1, 4)

Question. Let \( f(x) = \sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} \) and \( g(x) = \sec^2 x - \tan^2 x \). The two functions are equal over the set
(a) \( \phi \)
(b) R
(c) \( R - \left\{x | x = (2n + 1)\frac{\pi}{2}, n \in Z\right\} \)
(d) \( R - \{0\} \)
Answer: (c) \( R - \left\{x | x = (2n + 1)\frac{\pi}{2}, n \in Z\right\} \)

Question. \( f : R \rightarrow R \) defined by \( f(x) = \frac{x}{x^2 + 1}, \forall x \in R \) is
(a) one - one
(b) onto
(c) bijective
(d) neither one one nor onto
Answer: (d) neither one one nor onto

Question. If \( f : Z \rightarrow Z \) is such that \( f(x) = 6x - 11 \) then f is
(a) injective but not surjective
(b) surjective but not injective
(c) bijective
(d) neither injective nor surjective
Answer: (a) injective but not surjective

Question. \( f : N \rightarrow A \) Where A = {0,1} defined by \( f(x) = \begin{cases} 0 & \text{if } x \text{ is odd} \\ 1 & \text{if } x \text{ is even} \end{cases} \). Then f is
(a) one - one, onto
(b) one-one, into
(c) many-one, onto
(d) many-one, into
Answer: (c) many-one, onto

Question. \( f : (-\infty, \infty) \rightarrow (0, 1] \) defined by \( f(x) = \frac{1}{x^2 + 1} \) is
(a) one-one but not onto
(b) onto but not one-one
(c) bijective
(d) neither one-one nor onto
Answer: (b) onto but not one-one

Question. The function \( f : R \rightarrow R \) defined by \( f(x) = 4^x + 4^{|x|} \) is
(a) One - one and into
(b) Many - one and into
(c) One - one and onto
(d) Many-one and onto
Answer: (a) One - one and into

Number of functions :

Question. The number of one-one functions that can be defined from A = {4,8,12,16} to B is 5040, then n(B)=
(a) 7
(b) 8
(c) 9
(d) 10
Answer: (d) 10

Question. If A = {1,8,11,14,25} then the condition to define a surjection from A to B is
(a) n(A) + n(B) = 20
(b) n(A) < n(B)
(c) n(B) \( \le \) 5
(d) n(B) = 10
Answer: (c) n(B) \( \le \) 5

Question. If A = {1, 2, 3}, B = {1, 2} then the number of functions from A to B are
(a) 6
(b) 8
(c) 9
(d) 32
Answer: (b) 8

Question. The number of non-bijective mappings that can be defined from A = {1,2,7} to itself is
(a) 21
(b) 27
(c) 6
(d) 9
Answer: (a) 21

Question. Let \( A = \{1,2,3\} \) and \( B = \{a, b, c\} \). If l is number of funcitons from A to B and m is number of one-one functions from A to B, then
(a) l is 9
(b) m is 9
(c) l is 27
(d) m is 16
Answer: (c) l is 27

Question. The number of constant functions possible from R to B where B = {2,4,6,8,....24} are
(a) 24
(b) 12
(c) 8
(d) 6
Answer: (b) 12

Composite functions :

Question. The functions \( f : R \rightarrow R \), \( g : R \rightarrow R \) are defined as
\( f(x) = \begin{cases} 0 & \text{when } x \text{ is rational} \\ 1 & \text{when } x \text{ is irrational} \end{cases} \)
\( g(x) = \begin{cases} -1 & \text{when } x \text{ is rational} \\ 0 & \text{when } x \text{ is irrational} \end{cases} \)
then \( (fog)(\pi) + (gof)(e) = \) 
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (a) -1

Question. If \( f(x) = (a - x^n)^{1/n} \) then \( fof(x) \) is
(a) x
(b) a - x
(c) \( x^n \)
(d) \( x^{-1/n} \)
Answer: (a) x

Question. If \( f(x) = \frac{x}{\sqrt{1 + x^2}} \) then fofof (x) =
(a) \( \frac{x}{\sqrt{1 + 3x^2}} \)
(b) \( \frac{x}{\sqrt{1 - x^2}} \)
(c) \( \frac{2x}{\sqrt{1 + 2x^2}} \)
(d) \( \frac{x}{\sqrt{1 + x^2}} \)
Answer: (a) \( \frac{x}{\sqrt{1 + 3x^2}} \)

Inverse of a function:

Question. If \( f(x) = \frac{e^x + e^{-x}}{2} \) then the inverse of f(x) is
(a) \( \log_e(x + \sqrt{x^2 + 1}) \)
(b) \( \log_e \sqrt{x^2 + 1} \)
(c) \( \log_e(x + \sqrt{x^2 - 1}) \)
(d) \( \log_e(x - \sqrt{x^2 - 1}) \)
Answer: (c) \( \log_e(x + \sqrt{x^2 - 1}) \)

Question. If \( f : \{1, 2, 3,.....\} \rightarrow \{0, \pm 1, \pm 2,....\} \) is defined by \( f(n) = \begin{cases} n/2 & \text{if } n \text{ is even} \\ -\left(\frac{n-1}{2}\right) & \text{if } n \text{ is odd} \end{cases} \) then \( f^{-1}(-100) \) is
(a) 100
(b) 199
(c) 201
(d) 200
Answer: (c) 201

Question. If \( f(x) = \sin^{-1}\{3 - (x - 6)^4\}^{1/3} \) then \( f^{-1}(x) = \)
(a) \( 6 + \sqrt[4]{3 + \sin^3 x} \)
(b) \( 6 + \sqrt[4]{3 - \sin^3 x} \)
(c) \( 6 + \sqrt[4]{3 + \sin x} \)
(d) \( 6 + \sqrt[4]{3 - \sin x} \)
Answer: (b) \( 6 + \sqrt[4]{3 - \sin^3 x} \)

Real valued functions:

Question. f is defined by \( f(x) = \begin{cases} x^2, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases} \) then f(2)=
(a) 9
(b) 6
(c) 5
(d) not defined
Answer: (d) not defined

Question. If f(x) = ax+b, where a and b are integers, f(–1) = –5 and f(3)=3, then a and b are equal to
(a) a = –3, b = –1
(b) a = 2, b = –3
(c) a = 0, b = 2
(d) a = 2, b = 3
Answer: (b) a = 2, b = –3

Question. If \( f(x) = \begin{cases} x^2 + 1, & x \le 0 \\ 2x - 1, & 0 < x < 5 \\ 4x + 3, & x \ge 5 \end{cases} \) then \( \frac{f(-3) + f(2) + f(5)}{f(1)} = \)
(a) 28
(b) 36
(c) 26
(d) 34
Answer: (b) 36

Question. If f = {(-1,3), (0,2), (1,1)} then the range of \( f^2 - 1 \) is
(a) {0, 8}
(b) {0, 3, 8}
(c) {0, 1, 3}
(d) {0, 2, 8}
Answer: (b) {0, 3, 8}