Functions Questions (992)

If \(f(x+1) - f(x) = e^x\), then \(f(2) - f(0)\) equals

Ex. 9 Let $f(n)$ denote the square of the sum of the digits of natural number $n$, where $f^2(n)$ denotes $f(f(n))$, $f^3(n)$ denotes $f(f(f(n)))$ and so on. Then, the value of $\frac{f^{2017}(2011) - f^{2016}(2011)}{f^{2017}(2011) - f^{2018}(2011)}$ is

If graph of \(f(x)\) which is defined in \([-2, 2]\) is shown in the adjacent figure, then number of solution(s) of the equation \(f(x) = f^{-1}(x)\) is (are):

If \(f ( x )\) and \(g( x )\) are two functions such that \(f ( x ) = [ x ] + [ - x ]\) and \(g( x ) = \{ x \}\) for all \(x \in \mathbb{R}\), and \(h( x ) = f ( g( x ))\); then which of the following is incorrect? (where \([\cdot]\) denotes greatest integer function and \(\{\cdot\}\) denotes fractional part function)

Let f:[2, ∞) → [1, ∞) defined by f(x) = 2^{\frac{x^4 - 4x^2}{}}} be an invertible function. Find f^{-1}(x).

If $x = \log_b a = \log_a b = \dfrac{1}{2}\log_b c$ and $\log_c c = n(x)^{n+1}$, then the value of $n$ is

The domain of the function \(\sin^{-1}\left( \log_{2}\left( \frac{x}{3} \right) \right)\) is-

For all real number x, let f(x) = \frac{1}{1 - x^{2011}}. Find the number of real roots of the equation f(f(···(f(x))···)) = {−x} where f is applied 2013 times and {·} denotes fractional part function.

Suppose $f: \mathbb{R} \to (0,\infty)$ be a differentiable function such that $5f(x+y) = f(x) \cdot f(y)$, $\forall x, y \in \mathbb{R}$. If $f(3) = 320$, then $\displaystyle\sum_{n=0}^{5} f(n)$ is equal to:

Let \(f(x)=\sqrt{x-2}+\sqrt{4-x}\). Choose domain \(X\) and codomain \(Y\) so that \(f:X\to Y\) is bijective.

For \(f(x)=|x+3|-|x+1|-|x-1|+|x-3|\), which are correct?

Find the range of \(f(x) = \log_2\left(\dfrac{4}{\sqrt{x+2}+\sqrt{2-x}}\right)\)

For \(x \in \mathbb{R}\), \(x \neq 0\), \(x \neq 1\), let \(f_0(x) = \dfrac{1}{1-x}\) and \(f_{n+1}(x) = f_0(f_n(x))\), \(n = 0, 1, 2, \ldots\). Then the value of \(f_{100}(3) + f_1\left(\dfrac{2}{3}\right) + f_2\left(\dfrac{3}{2}\right)\) is equal to

Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is

A function $f:\mathbb{R}\to\mathbb{R}$ is defined as $f(x)=3x^2+1$. Then $f^{-1}(x)$ is

Consider the following statements:\(P\): Suman is brilliant.\(Q\): Suman is rich.\(R\): Suman is honest.The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as

Let f(x) = 2x - \left\lfloor \frac{x}{\pi} \right\rfloor and g(x) = \cos x, where \{ . \} denotes fractional part function, then period of gof(x) is -

Find the range of the following functions : (i) f(x) = \(\frac{x-1}{x+2}\)

Let $f:A\to B$ be a function defined by $f(x)=\sqrt{3}\sin x+\cos x+4$. If $f$ is both one-one and onto, then

Let f(x) = [x - 1] + \{x\}^{[x]}, x \in (1,3), then f^{-1}(x) is -

If functions f(x) and g(x) are defined on \mathbb{R} \to \mathbb{R} such that f(x) = \begin{cases} x + 3, & x \text{ rational} \\ \frac{x + \sqrt{5}}{-x}, & x \text{ irrational} \end{cases} then (f - g)(x) is-

The set of all possible values of parameter $a$ such that the equation $(1+a)\left(\dfrac{x^2}{1+x^2}\right)^2 - 3a\left(\dfrac{x^2}{1+x^2}\right) + 4a = 0$ has a real solution is

Period of f(x) = \{ x \} + \{ \frac{1}{3} x \} + \{ x + \frac{2}{3} \} is equal to (where \{ . \} denotes fractional part function)

Let f(x) = \frac{x}{1-x} and let \alpha be a real number. If x_0 = \alpha, x_1 = f(x_0), x_2 = f(x_1), \ldots and x_{2011} = -\frac{1}{2012} then the value of \alpha is

Range of function f(x) = \log_2 \left( \frac{4}{\sqrt{x+2}+\sqrt{2-x}} \right) is given by

If [x] and \{x\} denotes the greatest integer function less than or equal to x and fractional part function respectively, then the number of real x, satisfying the equation (x-2) [x] = \{x\} - 1, is

The number of integers lying in the domain of the function f(x) = \(\sqrt{\frac{5-2x}{x}}\) is -

The range of the function f(x) = \text{sgn}\left( \frac{\sin^{2} x + 2\sin x + 4}{\sin^{2} x + 2\sin x + 3} \right) is (where \text{sgn}(.) denotes signum function)-

The range of the function f(x) = \sqrt{4 - x^{2}} + \sqrt{x^{2} - 1} is

Suppose f(x,n) = \sum_{k=1}^{n} \log_x \left( \frac{k}{x} \right), then the value of x satisfying the equation f(x, 10) = f(x, 11) is

JM Q22.

The range of the function f : \(\mathbb{N} \to \mathbb{Z}\); f(x) = \((1-x)^{x-1}\), is -

The function $f(x)=2|x|+|x+2|-\big||x+2|-2|x|\big|$ has a local minima and a local maxima respectively at $x=$

The range of the function f(x) = \(e^{x}+e^{x}\), is -

Range of f(x) = \frac{\sec x + \tan x - 1}{\tan x - \sec x + 1}; x \in \left(0, \frac{\pi}{2}\right) is

A function f has domain [-1, 2] and range [0, 1]. The domain and range respectively of the function g defined by g(x) = 1 - f(x + 1) is

The period of the function \frac{\sin x + \sin 5x}{\cos x + \cos 5x} is -

Let $f(\theta) = \dfrac{1}{1+\cot^x\theta}$ (x-th power). Find $S = \displaystyle\sum_{\theta=1^\circ}^{89^\circ} f(\theta)$. Find $[S]$ (greatest integer).

The range of the function f(x) = \text{sgn}\left( \frac{\sin^{2} x + 2\sin x + 4}{\sin^{2} x + 2\sin x + 3} \right) is (where \text{sgn}(.) denotes signum function)-

The range of the function f(x) = \sqrt{4 - x^{2}} + \sqrt{x^{2} - 1} is

If 2f\left( x \right) - 3f\left( \frac{1}{x} \right) = x^2, x is not equal to zero, then f\left( 2 \right) is equal to-

If x^4 f\left( x \right) - \sqrt{1 - \sin 2\pi x} = |f\left( x \right)| - 2f\left( x \right), then f\left( -2 \right) equals

If f(x) = |x| then f^{-1}(x) equals-

The number of integers lying in the domain of the function f(x) = \(\sqrt{\frac{5-2x}{x}}\) is -

Which of the following function(s) have the same domain and range ?

Which of the following pair(s) of function(s) of function have same graphs?

Let f(x) = \{ x^{2}-3x + 4 ; x \} \{ x + 7 ; x ≥ 3 \} and g(x) = \{ x + 6 ; x \} \{ x^{2} + x + 2 ; x ≥ 4 \}, then which of the following is/are true -

Let $f(x)$ be any function. The graphs of $y = f(x-1)$ and $y = f(-x+1)$ are symmetric about the line:

If a function is defined by an implicit equation \(2|x| + 2|y| = 2\), then -

For each real x, let f(x) = \(\max\{x^{2}, x^{3}, x^{4}\}\), then f(x) is -