Class 11 Mathematics Functions MCQs Set 10

Composite functions:

Question. If \( f(x) = 2x + 1 \) and \( g(x) = x^2 + 1 \) then \( \text{go}(\text{fof})(2) = \)
(a) 112
(b) 122
(c) 12
(d) 124
Answer: (b) 122

Question. If \( f(x) = [x], g(x) = x - [x] \) then which of the following functions is a zero function
(a) \( (f + g)(x) \)
(b) \( (fg)(x) \)
(c) \( (f - g)(x) \)
(d) \( (fog)(x) \)
Answer: (d) \( (fog)(x) \)

Question. Let \( f(x) = \frac{Kx}{x+1} (x \neq -1) \) then the value of K for which \( (fof)(x) = x \) is
(a) 1
(b) -1
(c) 2
(d) \( \sqrt{2} \)
Answer: (b) -1

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

Question. If \( f : [-6, 6] \to R \) is defined by \( f(x) = x^2 - 3 \) for \( x \in R \) then
\( (fofof)(-1) + (fofof)(0) + (fofof)(1) = \)
(a) \( f(4\sqrt{2}) \)
(b) \( f(3\sqrt{2}) \)
(c) \( f(2\sqrt{2}) \)
(d) \( f(\sqrt{2}) \)
Answer: (a) \( f(4\sqrt{2}) \)

Question. If \( f(x) = (1-x)^{1/2} \) and \( g(x) = \ln(x) \) then the domain of (gof) (x) is
(a) \( (-\infty, 2) \)
(b) (-1, 1)
(c) \( (-\infty, 1] \)
(d) \( (-\infty, 1) \)
Answer: (d) \( (-\infty, 1) \)

Inverse of a function:

Question. If \( f : R^+ \to R \) such that \( f(x) = \log_5 x \) then \( f^{-1}(x) = \) 
(a) \( \log_x 10 \)
(b) \( 5^x \)
(c) \( 3^{-x} \)
(d) \( 3^{1/x} \)
Answer: (b) \( 5^x \)

Question. \( f : R \to R \) is a function defined by \( f(x) = 10x - 7 \). If \( g = f^{-1} \) then \( g(x) = \)
(a) \( \frac{1}{10x - 7} \)
(b) \( \frac{1}{10x + 7} \)
(c) \( \frac{x + 7}{10} \)
(d) \( \frac{x - 7}{10} \)
Answer: (c) \( \frac{x + 7}{10} \)

Question. \( f : [0, \infty) \to [4, \infty) \) is defined by \( f(x) = x^2 + 4 \) then \( f^{-1}(13) = \)
(a) 3
(b) 2
(c) 1
(d) 4
Answer: (a) 3

Question. If the function \( f : R \to R \) be such that \( f(x) = x - [x] \), where \( [x] \) denotes the greatest integer less than or equal to x then \( f^{-1}(x) \) is
(a) \( \frac{1}{x - [x]} \)
(b) \( [x] - x \)
(c) not defined
(d) \( x - [x] \)
Answer: (c) not defined

Real valued functions:

Question. Let \( g(x) \) be a function defined on \( [-1,1] \). If the area of the equilateral triangle with two of its vertices at \( (0,0) \) and \( (x, g(x)) \) is \( \sqrt{3}/4 \), then the function \( g(x) \) is
(a) \( g(x) = \pm\sqrt{1-x^2} \)
(b) \( g(x) = \sqrt{1-x^2} \)
(c) \( g(x) = -\sqrt{1+x^2} \)
(d) \( g(x) = \sqrt{1+x^2} \)
Answer: (b) \( g(x) = \sqrt{1-x^2} \)

Question. If \( f : R \rightarrow R \) is defined by \( f(x) = x - [x] - \frac{1}{2} \) for \( x \in R \), where \( [x] \) is the greatest integer not exceeding \( x \), then \( \left\{x \in R: f(x) = \frac{1}{2}\right\} = \)
(a) Z
(b) N
(c) \( \phi \)
(d) R
Answer: (c) \( \phi \)

Question. Suppose \( f : [-2, 2] \rightarrow R \) is defined by
\( f(x) = \begin{cases} -1 & \text{for } -2 \leq x \leq 0 \\ x - 1 & \text{for } 0 \leq x \leq 2 \end{cases} \)
then the \( \{x \in (-2, 2) : x \leq 0 \text{ and } f(|x|) = x \} = \)
(a) \( \{-1\} \)
(b) \( \{0\} \)
(c) \( \{-1/2\} \)
(d) \( \phi \)
Answer: (c) \( \{-1/2\} \)

Question. If \( |\sin x + \cos x| = |\sin x| + |\cos x| \), then \( x \) lies in
(a) 1st quadrant only
(b) 1st and 3rd quadrant only
(c) 2nd and 4th quadrant only
(d) 3rd and 4th quadrant only
Answer: (b) 1st and 3rd quadrant only

Even and odd functions:

Question. Let \( f(x) = \begin{cases} 4 & x < -1 \\ -4x & -1 \leq x \leq 0 \end{cases} \)
If \( f(x) \) is an even function on R then the definition of \( f(x) \) on \( (0, \infty) \) is
(a) \( f(x) = \begin{cases} 4x & 0 < x \leq 1 \\ 4 & x > 1 \end{cases} \)
(b) \( f(x) = \begin{cases} 4x & 0 < x \leq 1 \\ -4 & x > 1 \end{cases} \)
(c) \( f(x) = \begin{cases} 4 & 0 < x \leq 1 \\ 4x & x > 1 \end{cases} \)
(d) \( f(x) = \begin{cases} 4 & x < -1 \\ -4x & -1 \leq x \leq 0 \end{cases} \)
Answer: (a) \( f(x) = \begin{cases} 4x & 0 < x \leq 1 \\ 4 & x > 1 \end{cases} \)

Question. If \( f(x) = \begin{cases} x^2 \sin \frac{\pi x}{2} & |x| < 1 \\ x|x| & |x| \geq 1 \end{cases} \) then \( f(x) \) is
(a) an even funciton
(b) an odd function
(c) a periodic function
(d) neither odd nor even
Answer: (b) an odd function

Question. \( f(x) = \frac{\cos x}{\left[\frac{2x}{\pi}\right] + \frac{1}{2}} \), where \( x \) is not an integral multiple of \( \pi \) and \( [\cdot] \) denotes the greatest integer function is
(a) an odd function
(b) even function
(c) neither odd nor even
(d) both even and odd
Answer: (a) an odd function

Periodic functions:

Question. Which of the following function is not periodic
(a) \( \frac{2^x}{2^{[x]}} \)
(b) \( \sin^{-1}(\{x\}) \)
(c) \( \sin^{-1}(\sqrt{\cos x}) \)
(d) \( \sin^{-1}(\cos(x^2)) \)
Answer: (d) \( \sin^{-1}(\cos(x^2)) \)

Question. Let \( f(x) = nx + n - [nx + n] + \tan \frac{\pi x}{2} \), where \( [x] \) is the greatest integer \( \leq x \) and \( n \in N \). It is
(a) a periodic function of period 1
(b) a periodic function of period 4
(c) not periodic
(d) a periodic function of period 2
Answer: (d) a periodic function of period 2

Question. Let \( f(x) = x(2-x), 0 \leq x \leq 2 \). If the definition of \( f \) is extended over the set \( R - [0, 2] \) by \( f(x+2) = f(x) \) then \( f \) is a
(a) periodic function of period 1
(b) non periodic function
(c) periodic function of period 2
(d) periodic function of period 1/2
Answer: (c) periodic function of period 2

Question. If \( f \) is periodic, g is polynomial function and \( f(g(x)) \) is periodic and \( g(2) = 3, g(4) = 7 \) then \( g(6) \) is
(a) 13
(b) 15
(c) 11
(d) 21
Answer: (c) 11

Domain of the function:

Question. The domain of the function
\( f(x) = \frac{\sin^{-1}(x-3)}{\sqrt{9-x^2}} \) is 
(a) [2,3]
(b) [1,2)
(c) [1,2]
(d) [2,3)
Answer: (d) [2,3)

Question. The domain of \( f(x) = \text{Sin}^{-1}\left\{\log_3 \left(\frac{x^2}{3}\right)\right\} \) is
(a) \( (-\infty, 3] \)
(b) \( [3, \infty) \)
(c) \( [-3, -1] \cup [1, 3] \)
(d) \( (-9, -1) \cup (1, 9) \)
Answer: (c) \( [-3, -1] \cup [1, 3] \)

Question. The domain of \( f(x) = \log_x(9-x^2) \) is
(a) (-3,3)
(b) \( (0, \infty) \)
(c) \( (0, 1) \cup (1, \infty) \)
(d) \( (0, 1) \cup (1, 3) \)
Answer: (d) \( (0, 1) \cup (1, 3) \)

Question. The domain of \( f(x) = \frac{3}{4-x^2} + \log_{10}(x^3 - x) \) is 
(a) (1, 2)
(b) \( (-1, 0) \cup (1, 2) \)
(c) \( (1, 2) \cup (2, \infty) \)
(d) \( (-1, 0) \cup (1, 2) \cup (2, \infty) \)
Answer: (d) \( (-1, 0) \cup (1, 2) \cup (2, \infty) \)

Question. The domain of \( f(x) = \sqrt{1 - \sqrt{1 - \sqrt{1 - x^2}}} \) is
(a) [0,1]
(b) [-1,1]
(c) \( (-\infty, \infty) \)
(d) (-1,1)
Answer: (b) [-1,1]

Question. The domain of \( f(x) = \cos(\log x) \) is
(a) \( (-\infty, \infty) \)
(b) (-1,1)
(c) \( (0, \infty) \)
(d) \( (1, \infty) \)
Answer: (c) \( (0, \infty) \)

Question. The domain of \( f(x) = \cos^{-1}\left(\frac{2}{2 + \sin x}\right) \) contained in \( [0, 2\pi] \) is
(a) \( \left[0, \frac{\pi}{2}\right] \)
(b) \( \left[\frac{\pi}{2}, \pi\right] \)
(c) \( [0, \pi] \)
(d) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
Answer: (c) \( [0, \pi] \)

Question. \( \left\{x \in R : \left[x - [x]\right] = 5\right\} = \)
(a) R is the set of real numbers
(b) \( \phi \), the null set
(c) \( \{x \in R : x < 0\} \)
(d) \( \{x \in R : x \geq 0\} \)
Answer: (b) \( \phi \), the null set

Question. The domain of the function defined by
\( f(x) = {}^{(7-x)}P_{(x-3)} \) is
(a) {3,7}
(b) {3,4,5,6,7}
(c) {3,4,5}
(d) {1,2,3,4}
Answer: (c) {3,4,5}

Question. \( f : N \rightarrow N \) is defined as
\( f(n) = \begin{cases} 2, & n = 3k, k \in Z \\ 10 - n, & n = 3k + 1, k \in Z \\ 0, & n = 3k + 2, k \in Z \end{cases} \)
then \( \{n \in N : f(n) > 2\} = \) (EAM-04)
(a) {3, 6, 4}
(b) {1, 4, 7}
(c) {4, 7}
(d) {7}
Answer: (b) {1, 4, 7}

Question. The domain of the function
\( f(x) = \frac{1}{\sqrt{\{\sin x\} + \{\sin(\pi + x)\}}} \) where \( \{\cdot\} \) denotes the fractional part, is
(a) \( [0, \pi] \)
(b) \( (2n + 1)\frac{\pi}{2}, n \in Z \)
(c) \( (0, \pi) \)
(d) \( R - \left\{\frac{n\pi}{2}, n \in Z\right\} \)
Answer: (d) \( R - \left\{\frac{n\pi}{2}, n \in Z\right\} \)

Range of the function:

Question. If \( \alpha \in (0, \frac{\pi}{2}) \), then \( \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}} \) is always greater than or equal to (\( x \neq 0, -1 \))
(a) 2
(b) 1
(c) \( 2\tan \alpha \)
(d) \( 2\sec^2 \alpha \)
Answer: (c) \( 2\tan \alpha \)

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

Question. The range of \( f(x) = \text{Sin}^{-1} \left[\frac{1}{2} + x^2\right] \) is (\([\cdot]\) denotes greatest integer function)
(a) \( \left\{-\frac{\pi}{2}, 0, \frac{\pi}{2}\right\} \)
(b) \( \left\{0, \frac{\pi}{2}\right\} \)
(c) \( \left\{\frac{\pi}{2}\right\} \)
(d) \( \{0, \pi\} \)
Answer: (b) \( \left\{0, \frac{\pi}{2}\right\} \)

Question. If \( f : R \rightarrow R \) and \( g : R \rightarrow R \) defined by \( f(x) = |x| \) and \( g(x) = [x - 3] \) for \( x \in R \) (\([\cdot]\) is denotes greatest integer function) then \( \left\{g(f(x)) : \frac{-8}{5} < x < \frac{8}{5}\right\} = \)
(a) {0,1}
(b) {1,2}
(c) {-3,-2}
(d) {2,3}
Answer: (c) {-3,-2}

Question. The range of \( x^2 + 4y^2 + 9z^2 - 6yz - 3xz - 2xy \) is
(a) \( \phi \)
(b) R
(c) \( [0, \infty) \)
(d) \( (-\infty, 0) \)
Answer: (c) \( [0, \infty) \)

Question. The maximum possible domain and the corresponding range of \( f(x) = (-1)^x \) are
(a) \( D_f = R, R_f = [-1, 1] \)
(b) \( D_f = Z, R_f = \{1, -1\} \)
(c) \( D_f = Z, R_f = [-1, 1] \)
(d) \( D_f = R, R_f = \{-1, 1\} \)
Answer: (b) \( D_f = Z, R_f = \{1, -1\} \)

Question. The range of \( f(x) = \sin^{-1} x + \cos^{-1} x + \tan^{-1} x \) is
(a) \( (0, \pi) \)
(b) \( \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \)
(c) \( \left[\frac{-\pi}{4}, \frac{\pi}{4}\right] \)
(d) \( \left[0, \frac{3\pi}{4}\right] \)
Answer: (b) \( \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \)

Types of functions:

Question. Let A = [-1,1]= B then which of the following function from A to B is bijective function
(a) \( f(x) = \frac{x}{2} \)
(b) \( g(x) = |x| \)
(c) \( h(x) = x^2 \)
(d) \( k(x) = \sin \frac{\pi x}{2} \)
Answer: (d) \( k(x) = \sin \frac{\pi x}{2} \)

Question. If \( f : R \rightarrow C \) is defined by \( f(x) = e^{2ix} \) for \( x \in R \) then, f is (Where C denotes the set of all Complex numbers)
(a) One-one
(b) Onto
(c) One-one and Onto
(d) neither one-one nor Onto
Answer: (d) neither one-one nor Onto

Question. A function \( f : N \rightarrow Z \) defined by
\( f(n) = \begin{cases} \frac{n - 1}{2}, & \text{when } 'n' \text{ is odd} \\ \frac{-n}{2}, & \text{when } 'n' \text{ is even} \end{cases} \), is
(a) one-one but not onto
(b) onto but not one-one
(c) one-one onto
(d) neither one-one nor onto
Answer: (c) one-one onto

Question. M is the set of all \( 2 \times 2 \) real matrices. \( f : M \rightarrow R \) is defined by f(A)=det A for all A in M then f is
(a) one-one but not onto
(b) onto but not one-one
(c) neither one-one nor onto
(d) bijective
Answer: (b) onto but not one-one

Question. Let \( f : R - \{n\} \rightarrow R \) be a function defined by \( f(x) = \frac{x - m}{x - n} \) such that \( m \neq n \) then
(a) \( f \) is one one into function
(b) \( f \) is one one onto function
(c) \( f \) is many one into funciton
(d) \( f \) is many one onto funcion
Answer: (a) \( f \) is one one into function

Question. \( f(x) = \begin{cases} 0, & \text{if } x \text{ is rational} \\ x, & \text{if } x \text{ is irrational} \end{cases} \)
\( g(x) = \begin{cases} 0, & \text{if } x \text{ is irrational} \\ x, & \text{if } x \text{ is rational} \end{cases} \)
Then \( f - g \) is
(a) one-one and into
(b) neither one-one nor onto
(c) many one and onto
(d) one-one and onto
Answer: (d) one-one and onto

Composite functions:

Question. If \( f(x) \) and \( g(x) \) are two functions with \( g(x) = x - \frac{1}{x} \) and \( fog(x) = x^3 - \frac{1}{x^3} \), then \( f(x) = \)
(a) \( x^3 + 3x \)
(b) \( x^2 - \frac{1}{x^2} \)
(c) \( 1 + \frac{1}{x^2} \)
(d) \( 3x^2 + \frac{3}{x^4} \)
Answer: (a) \( x^3 + 3x \)

Question. Let \( f(x) = ax + b \) and \( g(x) = cx + d \), \( a \neq 0, c \neq 0 \). Assume \( a = 1, b = 2 \). If \( (fog)(x) = (gof)(x) \) for all x, what can you say about c and d
(a) c and d both arbitrary
(b) \( c = 1, d \) arbitrary
(c) c arbitrary, \( d = 1 \)
(d) \( c = 1, d = 1 \)
Answer: (b) \( c = 1, d \) arbitrary

Question. If \( f(x) = \sin^2 x \) and the composite functions \( g\{f(x)\} = |\sin x| \), then the function \( g(x) = \)
(a) \( \sqrt{x - 1} \)
(b) \( \sqrt{x} \)
(c) \( \sqrt{x + 1} \)
(d) \( -\sqrt{x} \)
Answer: (b) \( \sqrt{x} \)

Question. If \( f : R \rightarrow R \) and \( g : R \rightarrow R \) are given by \( f(x) = |x| \) and \( g(x) = [x] \) for each \( x \in R \), then \( \left\{x \in R : g(f(x)) \leq f(g(x))\right\} = \)
(a) \( Z \cup (-\infty, 0) \)
(b) \( (-\infty, 0) \)
(c) Z
(d) R
Answer: (d) R

Question. Let \( g : R \rightarrow R \) be given by \( g(x) = 3 + 4x \). If \( g^n(x) = gogo...og(x) \), and \( g^n(x) = A + Bx \) then A and B are
(a) \( 2^{n+1} - 1, 2^{n+1} \)
(b) \( 4^n - 1, 4^n \)
(c) \( 3^n, 3^n + 1 \)
(d) \( 5^n - 1, 5^n \)
Answer: (b) \( 4^n - 1, 4^n \)