Functions Questions (992)

Range of f(x) is -

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

The number of real numbers $x$ such that $\frac{x}{x+4} = \frac{5[x]-7}{7[x]-5}$ is ____. $[x]$ denotes greatest integer $\leq x$.

Range of g(f(x)) is -

Let $'a'$ be the real root of the equation $x^3 - 3x^2 + 5x - 17 = 0$ and $'b'$ be the real root of the equation $x^3 - 3x^2 + 5x + 11 = 0$. Then $a + b = $ ____.

Let $f(x) = 14^{\sin^2 x} + 14^{\cos^2 x}$. The number of integral values that $f(x)$ can take 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

For the function f(x) = \frac{e^{x} + 1}{e^{x} - 1}, if n(d) denotes the number of integers which are not in its domain and n(r) denotes the number of integers which are not in its range, then n(d) + n(r) is equal to -

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

Let f: A \to B be an onto function such that f(x) = \sqrt{-x - 2 - 2\sqrt{-3 - \sqrt{-x - 2 + 2\sqrt{-x - 3}}}, then set 'B' is-

Let f: \mathbb{R} \to \mathbb{R} be defined by f(x) = \ln(x + \sqrt{x^2 + 1}), then number of solutions of |f^{-1}(x)| = e^{|x|} is :-

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

The number of real solutions to the equation $3x - 7 = [x^2 - 3x + 2]$ is ____. $[x]$ denotes greatest integer $\leq x$.

Let f(x) = \(\sin^{6}(x) + \cos^{6}(x)\), then -

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

For the function f(x) = |x + 3| - |x + 1| - |x - 1| + |x - 3|, identify correct option(s)

A parabola of the form y = ax^2 + bx + c with a > 0 intersects the graph of f(x) = \frac{1}{x^2 - 4}. Number of possible distinct intersection(s) of these graph is

Let f(x) = \begin{cases} x^2 & ; 0 . Then :

Let $f(x) = (x+1)(x+2)(x+3)(x+4) + 5$ ; where $x \in [-6, 6]$. If the range of the function is $[a, b]$; where $a, b \in \mathbb{N}$, then find the value of $(a + b)$.

The number of integral values of x satisfying the inequality [x-5] + [x-3] + 2[x-5] + 2[x-3] (where [.] represents greatest integer function) 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

If f_1(x) = 2^{f_2(x)}, where f_2(x) = 2012^{f_3(x)}, where f_3(x) = \left( \frac{1}{2013} \right)^{f^{(x)}}, f_4(x) = \log_{2013}\log_{x}2012, then the range of f_1(x) is -

If $f(2x+1) = 4x^2 + 14x$, then find the sum of the squares of roots of the equation $f(x) = 0$.

Let $g(x) = \frac{e^x - e^{-x}}{2}$ and $g(f(x)) = x$, then evaluate $f\left(\frac{e^{22} - 1}{2e^{11}}\right)$.

Let $f(x) = \sin^3 x - \sin x \cos x + \cos^3 x$, then range of $f(x)$ is:

Let $f(x) = 1 + 2\cos x + 3\sin x$. If real numbers $a, b, c$ are such that $a f(x) + b f(-x) = 1$ holds for any $x \in \mathbb{R}$ then $\frac{b\cos c}{a} =$

The number of positive integers $x$ that satisfy $3^x = x^3 + 3x^2 + 2x + 1$ is:

$f(x)=\cot^{-1}\left(\dfrac{x+1}{x}\right)$ is defined for all real $x$ except

If g(x) = x^{2} - x + 1 and f(x) = \sqrt{\frac{1}{x} - x}, then -

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \log_{\sqrt{m}}\left\{\sqrt{2}(\sin x - \cos x) + m - 2\right\}$, for some $m$, such that the range of $f$ is $[0, 2]$. Then the value of $m$ is

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

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

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

Let the maximum value of expression $y=\dfrac{x^4-x^2}{x^6+2x^3-1}$ for $x>1$ be $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime natural numbers, then $p+q=$

Domain of $f(x)=\sqrt{\dfrac{2-|x|}{1-|x|}}$ is

Let $f(x)=\frac{1}{2}\begin{vmatrix}1&\tan x&1\\-\tan x&1&\tan x\\-1&-\tan x&1\end{vmatrix}+\begin{vmatrix}\cot\frac{\pi}{2}&\sec(x+\frac{\pi}{3})&\sec(x+\frac{\pi}{12})\\\csc(x-\frac{\pi}{6})&\sin2024\pi&e^{i2024\pi}\\\csc(x-\frac{5\pi}{12})&e^{2025\pi}&\tan(2025\pi)\end{vmatrix}$ and $g(x)=\sqrt{f(x)-1}+\sqrt{f(2025\pi/2-x)-1}$ on $(0,\pi/2)$. Let $m$ be minimum of $f(x)$ and $M$ minimum of $g(x)$. Range of $h(x)=(x-m)(x-M)$ on $[0,3]$ is

If \(S\) is the set of all real numbers. A relation \(R\) has been defined on \(S\) by \(aRb \Longleftrightarrow |a - b| \leq 1\), then \(R\) is:

Which of the following is the negation of the statement \(p \rightarrow (\sim p \vee \sim q)\)?

Let \(\mathbb{R}\) be the set of real numbers.Statement-1: \(A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : y - x \text{ is an integer}\}\) is an equivalence relation on \(\mathbb{R}\).Statement-2: \(B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x = \alpha y \text{ for some rational number } \alpha\}\) is an equivalence relation on \(\mathbb{R}\).

The contrapositive of the statement "I go to school if it does not rain" is:

Let \( f(x_1) = f(x_2) \), \( x_1, x_2 \in \mathbb{N} \) and \( f(x) = 4x + 3 \). Which of the following is true?

If $f(x)=\dfrac{4x+3}{6x-4}$, $x\neq\dfrac{2}{3}$ and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\left\{\dfrac{2}{3}\right\}\to\mathbb{R}-\left\{\dfrac{2}{3}\right\}$, then $(g\circ g\circ g)(4)$ is equal to

If the logarithms of two distinct positive numbers are equal to the corresponding numbers, then the base of the logarithm belongs to the interval

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:

Number of correct statements: I) $f,g:\mathbb{R}\to\mathbb{R}$ with $f(x)=x$ if rational, $0$ if irrational and $g(x)=0$ if rational, $x$ if irrational. Then $f-g$ is one-one and onto. II) $f(x)=\sin x+\cos ax$ is periodic only if $a$ is rational. III) If $f:A\to B$ and $g:B\to C$, $g\circ f:A\to C$ is one-one, then $f$ must be one-one.

Graph of $y=f(x)$ is given with $f(0)=3$, minimum at $x=1$, zero at $x=-1$ and $x=3$. Then graph of $y=\dfrac{-1}{|f(x)|}$ is best represented by:

If \(f(y) = \frac{x+1}{x-1}\), then \(f(y)\) is equal to

Given f : R → R be defined by \( f(x) = \dfrac{x}{1+x^2},\ x \in R \). Then f(x) ∈

Let $P(A)$ be defined as power set of set $A=\{1,2,3,\ldots,2025\}$. Let $R$ be a relation defined on $P(A)$ as $(B,C)\in R$ if $B$ is superset of $C$, then $R$ is

$f(x)=\left|x^2-x-6\right|/|x-3|$ at $x=3$. Which is true?