JEE Mathematics Functions MCQs Set 02

Question. Let \( f : R \to R \) be a function such that \( f(x) = x^3 - 6x^2 + 11x - 6 \). Then
(a) \( f \) is one-one and into
(b) \( f \) is many-one and into
(c) \( f \) is one-one and onto
(d) \( f \) is many-one and onto
Answer: (d) \( f \) is many-one and onto

Question. Let \( f : R \to R \) be a function such that \( f(x) = x^3 + x^2 + 3x + \sin x \). Then
(a) \( f \) is one-one and into
(b) \( f \) is one-one and onto
(c) \( f \) is many-one and into
(d) \( f \) is many-one and onto
Answer: (b) \( f \) is one-one and onto

Question. The function \( f : R \to R \) defined by \( f(x) = 6^x + 6^{|x|} \) is
(a) one-one and onto
(b) many-one and onto
(c) one-one and into
(d) many-one and into
Answer: (c) one-one and into

Question. If the real-valued function \( f(x) = px + \sin x \) is a bijective function then the set of possible value of \( p \in R \) is
(a) \( R - \{0\} \)
(b) \( R \)
(c) \( (0, +\infty) \)
(d) none of the options
Answer: (a) \( R - \{0\} \)

Question. Let \( f(x) = 2x + |\cos x| \). Then \( f \) is
(a) one-one and into
(b) one-one and onto
(c) many-one and into
(d) many-one and onto
Answer: (b) one-one and onto

Question. Let \( f \) be a function from \( R \) to \( R \) given by \( f(x) = \frac{x^2 - 4}{x^2 + 1} \). Then \( f(x) \) is
(a) one-one and into
(b) one-one and onto
(c) many-one and into
(d) many-one and onto
Answer: (c) many-one and into

Question. Let \( f : R \to A = \{y | 0 \leq y < \frac{\pi}{2}\} \) be a function such that \( f(x) = \tan^{-1}(x^2 + x + k) \), where \( k \) is a constant. The value of \( k \) for which \( f \) is an onto function, is
(a) 1
(b) 0
(c) \( \frac{1}{4} \)
(d) none of the options
Answer: (c) \( \frac{1}{4} \)

Question. \( f(x) = x + \sqrt{x^2} \) is a function from \( R \to R \). Then \( f(x) \) is
(a) injective
(b) surjective
(c) bijective
(d) none of the options
Answer: (d) none of the options

Question. Which of the following is an even function? Here \( [.] \) denotes the greatest integer function and \( f \) is any function.
(a) \( [x] - x \)
(b) \( f(x) - f(-x) \)
(c) \( e^{3-2x} . \tan^2 x \)
(d) \( f(x) + f(-x) \)
Answer: (d) \( f(x) + f(-x) \)

Question. Let \( f(x) = |x - 2| + |x - 3| + |x - 4| \) and \( g(x) = f(x + 1) \). Then
(a) \( g(x) \) is an even function
(b) \( g(x) \) is an odd function
(c) \( g(x) \) is neither even nor odd
(d) \( g(x) \) is periodic
Answer: (c) \( g(x) \) is neither even nor odd

Question. \( f(x) = \log_{10} (x + \sqrt{x^2 + 1}) \) is
(a) an odd function
(b) a periodic function
(c) an even function
(d) none of the options
Answer: (a) an odd function

Question. A function whose graph is symmetrical about the y-axis is given by
(a) \( f(x) = \log_e (x + \sqrt{x^2 + 1}) \)
(b) \( f(x + y) = f(x) + f(y) \) for all \( x, y \in R \)
(c) \( f(x) = \cos x + \sin x \)
(d) none of the options
Answer: (d) none of the options

Question. A function whose graph is symmetrical about the origin is given by
(a) \( f(x) = e^x + e^{-x} \)
(b) \( f(x) = \log_e x \)
(c) \( f(x + y) = f(x) + f(y) \)
(d) none of the options
Answer: (c) \( f(x + y) = f(x) + f(y) \)

Question. Let \( f(x) = 4, x < -1 \)
\( -4x, -1 \leq x \leq 0 \).
If \( f(x) \) is an even function in \( R \) then the definition of \( f(x) \) in \( (0, +\infty) \) is
(a) \( f(x) = 4x, 0 < x \leq 1; \quad 4, x > 1 \)
(b) \( f(x) = 4x, 0 < x \leq 1; \quad -4, x > 1 \)
(c) \( f(x) = 4, 0 < x \leq 1; \quad 4x, x > 1 \)
(d) none of the options
Answer: (a) \( f(x) = 4x, 0 < x \leq 1; \quad 4, x > 1 \)

Question. If \( f(x) = x^2 \sin \frac{\pi x}{2}, |x| < 1 \)
\( x |x|, |x| \geq 1 \) then \( f(x) \) is
(a) an even function
(b) an odd function
(c) a periodic function
(d) none of the options
Answer: (b) an odd function

Question. The period of the function \( f(x) = \left| \sin \frac{x}{2} \right| + | \cos x | \) is
(a) \( 2\pi \)
(b) \( \pi \)
(c) \( 4\pi \)
(d) none of the options
Answer: (a) \( 2\pi \)

Question. If \( f(x) \) is a periodic function of the period \( k \) then \( f(kx + a) \), where \( a \) is a constant, is a periodic function of the period
(a) \( k \)
(b) 1
(c) \( \frac{k}{a} \)
(d) none of the options
Answer: (b) 1

Question. The period of the function \( f(x) = 4\cos(2x + 3) \) is
(a) \( 2\pi \)
(b) \( \frac{\pi}{2} \)
(c) \( \pi \)
(d) none of the options
Answer: (c) \( \pi \)

Question. The period of the function \( f(x) = 3\sin \frac{\pi x}{3} + 4\cos \frac{\pi x}{4} \) is
(a) 6
(b) 24
(c) 8
(d) \( 2\pi \)
Answer: (b) 24

Question. Let \( f(x) = \cos \sqrt{px} \), where \( p = [a] = \) the greatest integer less than or equal to \( a \). If the period of \( f(x) \) is \( \pi \) then
(a) \( a \in [4, 5] \)
(b) \( a = 4, 5 \)
(c) \( a \in [4, 5) \)
(d) none of the options
Answer: (c) \( a \in [4, 5) \)

Question. Let \( f(x) = \cos 3x + \sin \sqrt{3}x \). Then \( f(x) \) is
(a) a periodic function of period \( 2\pi \)
(b) a periodic function of period \( \sqrt{3}\pi \)
(c) not a periodic function
(d) none of the options
Answer: (c) not a periodic function

Question. The function \( f(x) = \sin \frac{\pi x}{n!} - \cos \frac{\pi x}{(n + 1)!} \) is
(a) not periodic
(b) periodic, with period \( 2(n!) \)
(c) periodic, with period \( (n + 1) \)
(d) none of the options
Answer: (d) none of the options

Question. The function \( f(x) = x - [x] + \cos x \), where \( [x] = \) the greatest integer less than or equal to \( x \), is a
(a) periodic function of indeterminate period
(b) periodic function of period \( 2\pi \)
(c) nonperiodic function
(d) periodic function of period 1
Answer: (c) nonperiodic function

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 \( f(x + 2) = f(x) \) the \( f \) is a
(a) periodic function of period 1
(b) nonperiodic function
(c) periodic function of period 2
(d) periodic function of period \( \frac{1}{2} \)
Answer: (c) periodic function of period 2

Question. If \( f(x) = \sin^2 x + \sin^2 \left(x + \frac{\pi}{3}\right) + \cos x \cos \left(x + \frac{\pi}{3}\right) \) and \( g\left(\frac{5}{4}\right) = 1 \) then \( (gof)(x) \) is
(a) a polynomial of the first degree in \( \sin x, \cos x \)
(b) a constant function
(c) a polynomial of the second degree in \( \sin x, \cos x \)
(d) none of the options
Answer: (b) a constant function

Question. If \( f(x) = x^n, n \in N \) and \( (gof)(x) = ng(x) \) then \( g(x) \) can be
(a) \( n |x| \)
(b) \( 3 . \sqrt[3]{x} \)
(c) \( e^x \)
(d) \( \log |x| \)
Answer: (d) \( \log |x| \)

Question. If \( g|f(x)| = |\sin x| \) and \( f\{g(x)\} = (\sin \sqrt{x})^2 \) then
(a) \( f(x) = \sin^2 x, g(x) = \sqrt{x} \)
(b) \( f(x) = \sin x, g(x) = | x | \)
(c) \( f(x) = x^2, g(x) = \sin \sqrt{x} \)
(d) \( f \) and \( g \) cannot be determined
Answer: (a) \( f(x) = \sin^2 x, g(x) = \sqrt{x} \)

Question. If \( f(x) = \frac{1}{1 - x}, x \neq 0, 1, \) then the graph of the function \( y = f\{f(f(x))\}, x > 1, \) is
(a) a circle
(b) an ellipse
(c) a straight line
(d) a pair of straight lines
Answer: (c) a straight line

Question. If \( f(x) \) is a polynomial function of the second degree such that \( f(-3) = 6, f(0) = 6 \) and \( f(2) = 11 \) then the graph of the function \( f(x) \) cuts the ordinate \( x = 1 \) at the point
(a) (1, 8)
(b) (1, 4)
(c) (1, -2)
(d) none of the options
Answer: (a) (1, 8)

Question. Let \( f(x) \) be a function whose domain is [-5, 7]. Let \( g(x) = |2x + 5| \). Then the domain of \( (fog)(x) \) is
(a) [-5, 1]
(b) [-4, 0]
(c) [-6, 1]
(d) none of the options
Answer: (c) [-6, 1]

Question. Let \( f : (-\infty, 1] \to (-\infty, 1] \) such that \( f(x) = x(2 - x) \). Then \( f^{-1}(x) \) is
(a) \( 1 + \sqrt{1 - x} \)
(b) \( 1 - \sqrt{1 - x} \)
(c) \( \sqrt{1 - x} \)
(d) none of the options
Answer: (b) \( 1 - \sqrt{1 - x} \)

Question. If \( f(x) = 3x - 5 \) then \( f^{-1}(x) \)
(a) is given by \( \frac{1}{3x - 5} \)
(b) is given by \( \frac{x + 5}{3} \)
(c) does not exist because \( f \) is not one-one
(d) does not exist because \( f \) is not onto
Answer: (b) is given by \( \frac{x + 5}{3} \)

Question. If the function \( f: [1, +\infty) \to [1, +\infty) \) is defined by \( f(x) = 2^{x(x-1)} \) then \( f^{-1}(x) \) is
(a) \( \left(\frac{1}{2}\right)^{x(x-1)} \)
(b) \( \frac{1}{2}(1 + \sqrt{1 + 4\log_2 x}) \)
(c) \( \frac{1}{2}(1 - \sqrt{1 + 4\log_2 x}) \)
(d) not defined
Answer: (b) \( \frac{1}{2}(1 + \sqrt{1 + 4\log_2 x}) \)

Question. If the function \( f : R \to R \) be such that \( f(x) = x - [x] \), where \( [y] \) denotes the greatest integer less than or equal to \( y \), then \( f^{-1}(x) \) is
(a) \( \frac{1}{x - [x]} \)
(b) \( x - [x] \)
(c) not defined
(d) none of the options
Answer: (c) not defined

Question. The inverse function of the function \( f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) is
(a) \( \frac{1}{2}\log \frac{1 + x}{1 - x} \)
(b) \( \frac{1}{2}\log \frac{2 + x}{2 - x} \)
(c) \( \frac{1}{2}\log \frac{1 - x}{1 + x} \)
(d) none of the options
Answer: (a) \( \frac{1}{2}\log \frac{1 + x}{1 - x} \)

Question. If \( f(x + y, x - y) = xy \) then the arithmetic mean of \( f(x, y) \) and \( f(y, x) \) is
(a) \( x \)
(b) \( y \)
(c) 0
(d) none of the options
Answer: (c) 0

Question. The graph of the function \( y = f(x) \) is symmetrical about the line \( x = 2 \). Then
(a) \( f(x + 2) = f(x - 2) \)
(b) \( f(2 + x) = f(2 - x) \)
(c) \( f(x) = f(-x) \)
(d) none of the options
Answer: (b) \( f(2 + x) = f(2 - x) \)

Choose the correct options. One or more options may be correct.

Question. If \( e^x + e^{f(x)} = e \) then for \( f(x) \)
(a) domain = \( (-\infty, 1) \)
(b) range = \( (-\infty, 1) \)
(c) domain = \( (-\infty, 0] \)
(d) range = \( (-\infty, 1] \)
Answer: (a) domain = \( (-\infty, 1) \), (b) range = \( (-\infty, 1) \)

Question. If \( f(x) \) is an odd function then
(a) \( \frac{f(-x) + f(x)}{2} \) is an even function
(b) \( [|f(x)| + 1] \) is even, where \( [x] = \) the greatest integer \( \leq x \)
(c) \( \frac{f(x) - f(-x)}{2} \) is neither eve nor odd
(d) none of the options
Answer: (a) \( \frac{f(-x) + f(x)}{2} \) is an even function, (b) \( [|f(x)| + 1] \) is even, where \( [x] = \) the greatest integer \( \leq x \)

Question. Let \( f(x) = \sec^{-1}[1 + \cos^2 x] \) where \( [.] \) denotes the greatest integer function. Then
(a) the domain of \( f \) is \( R \)
(b) the domain of \( f \) is [1, 2]
(c) the range of \( f \) is [1, 2]
(d) the range of \( f \) is \( \{\sec^{-1} 1, \sec^{-1} 2\} \)
Answer: (a) the domain of \( f \) is \( R \), (d) the range of \( f \) is \( \{\sec^{-1} 1, \sec^{-1} 2\} \)

Question. If \( f(x) \) and \( g(x) \) are two functions of \( x \) such that \( f(x) + g(x) = e^x \) and \( g(x) - f(x) = e^{-x} \) then
(a) \( f(x) \) is an odd function
(b) \( g(x) \) is an odd function
(c) \( f(x) \) is an even function
(d) \( g(x) \) is an even function
Answer: (b) \( g(x) \) is an odd function, (c) \( f(x) \) is an even function

Question. Let \( f(x) = \sqrt{4\cos \sqrt{x^2 - \frac{\pi^2}{9}}} \). Then
(a) the domain of \( f \) is \( [\frac{\pi}{3}, +\infty) \)
(b) the range of \( f \) is [-1, 1]
(c) the domain of \( f \) is \( (-\infty, -\frac{\pi}{3}] \cup [\frac{\pi}{3}, +\infty) \)
(d) the range of \( f \) is [-4, 4]
Answer: (c) the domain of \( f \) is \( (-\infty, -\frac{\pi}{3}] \cup [\frac{\pi}{3}, +\infty) \), (d) the range of \( f \) is [-4, 4]

Question. Let \( f(x + y) = f(x) + f(y) \) for all \( x, y \in R \). Then
(a) \( f(x) \) is an even function
(b) \( f(x) \) is an odd function
(c) \( f(0) = 0 \)
(d) \( f(n) = nf(1), n \in N \)
Answer: (b) \( f(x) \) is an odd function, (c) \( f(0) = 0 \), (d) \( f(n) = nf(1), n \in N \)

Question. Let \( f(x) = [x]^2 + [x + 1] - 3 \), where \( [x] = \) the greatest integer \( \leq x \). Then
(a) \( f(x) \) is a many-one and into function
(b) \( f(x) = 0 \) for infinite number of values of \( x \)
(c) \( f(x) = 0 \) for only two real values
(d) none of the options
Answer: (a) \( f(x) \) is a many-one and into function, (b) \( f(x) = 0 \) for infinite number of values of \( x \)

Question. Let \( f \) and \( g \) be functions from the interval \( [0, \infty) \) to the interval \( [0, \infty) \) \( f \) being an increasing function and \( g \) being a decreasing function. If \( f\{g(0)\} = 0 \) then
(a) \( f\{g(x)\} \geq f\{g(0)\} \)
(b) \( g\{f(x)\} \leq g\{f(0)\} \)
(c) \( f\{g(2)\} = 0 \)
(d) none of the options
Answer: (b) \( g\{f(x)\} \leq g\{f(0)\} \), (c) \( f\{g(2)\} = 0 \)