Functions Questions - JEE

The domain of the definition of the function f(x) = 1/(4 − x2) + log₁₀(x3 − x) is

Let f(x) = 8x + 3 and g(x) = x² + 4. Find (g∘f)(2).

Find $f(x) = \frac{1}{2}|x+2| - 1\bigg|\frac{1}{2}(x-2)\bigg| + 1$. Hence find the range of $f(x)$.

100. If $f(x) = x^3 - 3x + 1$, then minimum number of real roots of $f(f(x)) = 0$ is:

Domain of the function f(x), if 3^x + 3^{f(x)} = \text{minimum of } F(t), where F(t) = \min\{2t^3 - 15t^2 + 36t - 25, 2|\sin t|\} is

Let f(x) = 7x² + x - 8 and g(x) = |x|. Find (g∘f)(0) + (f∘g)(-3).

Let $f(x) = \frac{1}{1-x}$, then {$f \circ (f \circ f)$}(100) is equal to

Let f : \mathbb{R} \to \mathbb{R} and g : \mathbb{R} \to \mathbb{R} be two one-one and onto functions, such that they are the mirror images of each other about the line y = a. If h(x) = f(x) + g(x), then h(x) is

Let max{|x + y|, |x - y|} = 1 and y = x - [x] be two equations which x and y satisfy, then the number of ordered pairs (x, y) is

If the relation R : A → B, where A = {1, 2, 3, 4} and B = {1, 3, 5} is defined by R = {(x, y) : x y, x ∈ A, y ∈ B}, then RoR−1 is

136. Let $f:R\to R$ be given as \(f(x)=\begin{cases}2x+\alpha^2, & x\geq 2\\ \dfrac{\alpha x}{2}+10, & x

50. Let $f$ be an invertible function from $R \to R$ satisfying the equation \[f^3(x) - (x^3 + 2)f^2(x) + (2x^3 + 1)f(x) - x^3 = 0.\] Then the value of $f'(8) \times (f^{-1})'(8)$ is:

The domain of $f(g(x))$ is

For any x = a ≥ 5, f(a) = $\sqrt{a-5}$ ≥ 0. What is the range of the function?

Range of the function f(x) = log2(2 − log2(16 sin2x + 1)) is:

The function $f(x) = \dfrac{\sqrt{x^2 + kx + 1}}{x^2 - k}$ is continuous for all real $x$. Find the range of $k$.

The value of a and b for which |e|x−b| − a| = 2 has four distinct solutions, are:

If $f(x) + 2f\left(\dfrac{1}{x}\right) = 3x$, $x \neq 0$ and $S = \{x \in \mathbb{R} : f(x) = f(-x)\}$, then $S$

Let $\sum_{k=1}^{10} f(a+k) = 16(2^{10}-1)$, where the function $f$ satisfies $f(x+y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2$. Then the natural number $a$ is ______.

Let $f(x)=2\tan^{-1}x$ and $g(x)$ differentiable with $g\!\left(\dfrac{x+2y}{3}\right)=\dfrac{g(x)+2g(y)}{3}$, $g'(0)=1$, $g(0)=2$. Number of integers $x$ in $(-10,20)$ satisfying $f^2(g(x))-5f(g(x))+4>0$ is

135. The function $f:[0,\infty)\to[0,\infty)$ defined by $f(x)=\dfrac{2x}{1+2x}$ 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).

Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A → B | 2 ∈ f(A) and f is not one-one}

The function $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x - 5\left[\dfrac{x}{5}\right]$, where $\mathbb{N}$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to $x$, is

787. Graph of a function $y = f(x)$ is shown. If $g(x) = |f(|x|)|$, then find number of solution(s) of the equation $g(g(x)) = \text{sgn}(x^2 - (k+1)x + (k^2 + 1))$, $k \in R$.[Note:] sgn$(y)$ denotes the signum function of $y$.

The function $ f(x) = \sqrt{x-5} $ is defined for $ x \geq 5 $. What is the domain of $ f $?

If $f(x) = \sin\left[\log\left(\frac{4-x^2}{1-x}\right)\right]; x \in \mathbb{R}$, then range of $f(x)$ is given by:

Given $ f_1(x) = \dfrac{1}{x} $, $ f_2(x) = 1 - x $, $ f_3(x) = \dfrac{1}{1-x} $ and $ (f_2 \circ J \circ f_1)(x) = f_3(x) $. Then $ J(x) $ equals:

For a real number x, let [x] denote the greatest integer less than or equal to x. Let f: ℝ → ℝ be defined by f(x) = 2x + [x] + sin x cos x. Then f is:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \dfrac{|x|-1}{|x|+1}$, then $f$ is

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

Consider the following relations. $R = \{(x, y) | x, y \text{ are real numbers and } x = wy \text{ for some rational number } w\}$ and $S = \left\{\frac{m}{p}; \frac{n}{q} \mid m, n, p, q \text{ are integers such that } n, q > 0 \text{ and } qm = pn\right\}$, then

Find the natural number $a$ for which $\displaystyle\sum_{k=1}^{n} f(a+k) = 16(2^n - 1)$, where the function $f$ satisfies $f(x+y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2$.

Let f and g be two differentiable functions on R such that f'(x) > 0 and g'(x) > 0 for all x ∈ R. Then for all x:Which of the following is true?

For $ x \in \mathbb{R} - \{0, 1\} $, let f_1(x) = \frac{1}{x}, f_2(x) = 1 - x and f_3(x) = \frac{1}{1 - x} \) be three given functions. If a function, J(x) satisfies $ f_2 \circ f_1 \circ f(x) = f_3(x) $ then J(x) is equal to :-

Given f(xy) = f(x) · f(y), for all $x, y \in [0,1]$. Since $f(0) \neq 0$, then $f(0) = 1$. So $f(x) = 1$. Find $y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)$.

Let $f(x)$ is continuous function with range $[-1, 1]$ and $f(x)$ is defined $\forall x \in \mathbb{R}$. If $g(x) = \frac{e^{f(x)} - e^{|f(x)|}}{e^{f(x)} + e^{|f(x)|}}$, then range of $g(x)$ is:

Find the number of integers in the domain of $f(x)=\dfrac{1}{\sqrt{\ln\cos^{-1}x}}$.

Ex. 1 Let $f(x) = \frac{a_{2k}x^{2k} + a_{2k-1}x^{2k-1} + \ldots + a_1x + a_0}{b_{2k}x^{2k} + b_{2k-1}x^{2k-1} + \ldots + b_1x + b_0}$ where $k$ is a positive integer, $a_i, b_i \in \mathbb{R}$ and $a_{2k} \neq 0$, $b_{2k} \neq 0$ such that $b_{2k}x^{2k} + b_{2k-1}x^{2k-1} + \ldots + b_1x + b_0 = 0$ has no real roots, then

The domain of the function f(x) = \sin^{-1}\left(\frac{1}{2} - \frac{1}{|x-1|} + \sin^{-1}x + \sin x - 1\right) is

Let f : \mathbb{R} \to \mathbb{R} defined by f(x) = x^3 + ax^2 + 3x + 100, then the values of a for which f is a one-one function, is

The domain of the function $f(x) = \dfrac{\sin^{-1}(x-3)}{\sqrt{9-x^2}}$ is

If $f:\{1,2,3,4\}\to\{1,2,3,4\}$ is a function such that $|f(\alpha)-\alpha|\leq 1$ for $\alpha\in\{1,2,3,4\}$, then total number of such functions is

If $p$ and $q$ are +ve integers, $f$ is a function defined for +ve numbers and attains only positive values such that $f(xf(y)) = x^p y^q$, then

Find the number of elements contained in the range of the function f(x) = \left\lfloor \frac{x}{6} \right\rfloor \left\lfloor \frac{−6}{x} \right\rfloor for x ∈ (0, 30] (where [·] denotes greatest integer function)

Let $f(x) = \sin\!\left(\dfrac{\pi}{6}\sin\!\left(\dfrac{\pi}{2}\sin x\right)\right)$ for all $x \in R$. Then the range of $f(x)$ is:

If $A = \{1, 2, 3, 4\}$ and $f: A \to A$, then total number of invertible functions $f$ such that $f(2) \neq 2$, $f(4) \neq 4$, $f(1) = 1$ is equal to:

If $y = 2^{x(x-1)}$ and $x \geq \dfrac{1}{2}$, find $f^{-1}(x)$.

If f : ℝ → $\left[-1, 1\right]$, f(x) = $\sin\left(\tan^{-1}\left(\frac{x^2 - a}{x^2 + 1}\right)\right)$ is an onto function, the set of values of 'a' is

Consider the function $f : \mathbb{R} - \{1\} \to \mathbb{R} - \{2\}$ given by $f(x) = \frac{2x}{x-1}$. Then:

Functions Questions (992)