Functions PYQs - Last 5 Years

If a differentiable function satisfies $(x-y) f(x+y)-(x+y) f(x-y)=2\left(x^2 y-y^3\right) \forall x, y \in \mathbb{R}$ and $f(I)=2$, then

If the domain of $f(x)$ is $(0,1)$, then the domain of $y=f\left(e^x\right)+f(\ln |x|)$ is

Let $A=[a, \infty)$ denotes the domain, then $f:(a, \infty) \rightarrow B$, which is defined by $f(x)=2 x^3-3 x^2+6$ will have an inverse for the smallest real value of ' $a$ ' if

If $f$ be a real valued function defined for all real numbers $x$ such that for some fixed $a>0$, it satisfies $f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^2} \forall x$, then $f(x)$ is periodic with period

Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in(0,90$ ); (where [.] denotes the greatest integer function) is

If $f(x)=\frac{3 x-4}{2 x-3}$, then $f(f(f(x)))$ will be

If $f(x)$ and $g(x)$ are two polynomials such that $\phi(x)=f\left(x^3\right)+x g\left(x^3\right)$ is divisible by $x^2+x+1$, then

Let $u+v+w=3, u, v, w \in \mathbb{R}$ and $f(x)=u x^2+v x+w$ be such that $f(x+y)=f(x)+f(y)+x y$, $\forall x, y \in \mathbb{R}$. Then $f(1)$ is equal to

If $g(f(x))=|\sin x|$ and $f(g(x))=(\sin \sqrt{x})^2$, then

Choose the correct statement :

The function \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(\mathrm{f}(x)=\mathrm{e}^x+\mathrm{e}^{-x}\) is :

The equation \(2^x+5^x=3^x+4^x\) has

For every real number \(x \neq-1\), let \(\mathrm{f}(x)=\frac{x}{x+1}\).
Write \(\mathrm{f}_1(x)=\mathrm{f}(x)\) & for \(\mathrm{n} \geq 2, \mathrm{f}_{\mathrm{n}}(x)=\mathrm{f}\left(\mathrm{f}_{\mathrm{n}-1}(x)\right)\). Then $$\mathrm{f}_...

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(\mathrm{f}(x)=\frac{\mathrm{e}^{|x|}-\mathrm{e}^{-x}}{\mathrm{e}^x+\mathrm{e}^{-x}}\), then

In the interval \(( - 2\pi ,0)\), the function \(f(x) = \sin \left( {{1 \over {{x^3}}}} \right)\).

Let p(x) be a polynomial with real co-efficient, p(0) = 1 and p'(x) > 0 for all x \(\in\) R. Then

Let \(f(x) = {x^2} + x\sin x - \cos x\). Then

\(f:X \to R,X = \{ x|0 < x < 1\}\) is defined as \(f(x) = {{2x - 1} \over {1 - |2x - 1|}}\). Then

The maximum value of \(f(x) = {e^{\sin x}} + {e^{\cos x}};x \in R\) is

Let \(f(n) = {2^{n + 1}}\), \(g(n) = 1 + (n + 1){2^n}\) for all \(n \in N\). Then

Let \(f(x) = {(x - 2)^{17}}{(x + 5)^{24}}\). Then

Domain of \(y = \sqrt {{{\log }_{10}}{{3x - {x^2}} \over 2}}\) is