Functions

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

Let f and g be periodic functions with the periods T1 and T2 respectively. Then f + g is

Given that f : S \(\to\) R is said to have a fixed point at c of S if f(c) = c. Let f : [1, \(\infty\)) \(\to\) R be defined by f(x) = 1 + \(\sqrt x\). Then

Consider the functions f1(x) = x, f2(x) = 2 + loge x, x > 0. The graphs of the functions intersect

f(x) is real valued function such that 2f(x) + 3f(\(-\)x) = 15 \(-\) 4x for all x\(\in\)R. Then f(2) =

Let f : R \(\to\) R be given by f(x) = | x2 \(-\) 1 |, x\(\in\)R. Then,

Let \(f(x) = 1 - \sqrt {({x^2})}\), where the square root is to be taken positive, then

The domain of \(f(x) = \sqrt {\left( {{1 \over {\sqrt x }} - \sqrt {x + 1} } \right)}\) is

Let \(A = \{ x \in R: - 1 \le x \le 1\}\) and \(f:A \to A\) be a mapping defined by \(f(x) = x\left| x \right|\). Then f is

Let \(f(x) = \sqrt {{x^2} - 3x + 2}\) and \(g(x) = \sqrt x\) be two given functions. If S be the domain of fog and T be the domain of gof, then

Let a > b > 0 and I(n) = a1/n \(-\) b1/n, J(n) = (a \(-\) b)1/n for all n \(\ge\) 2, then

Consider the function f(x) = cos x2. Then,

Let \(f:R \to R\) be defined by \(f(x) = {x^2} - {{{x^2}} \over {1 + {x^2}}}\) for all \(x \in R\). Then,

For 0 \(\le\) p \(\le\) 1 and for any positive a, b; let I(p) = (a + b)p, J(p) = ap + bp, then

The domain of definition of \(f(x) = \sqrt {{{1 - |x|} \over {2 - |x|}}}\) is

If f : R \(\to\) R be defined by f (x) = ex and g : R \(\to\) R be defined by g(x) = x2. The mapping gof : R \(\to\) R be defined by (gof) (x) = g[f(x)] \(\forall\)x\(\in\)R. Then,

Consider the function \(y = {\log _a}(x + \sqrt {{x^2} + 1} ),a > 0,a \ne 1\). The inverse of the function

Let \(f(x) = {x^{13}} + {x^{11}} + {x^9} + {x^7} + {x^5} + {x^3} + x + 19\). Then, f(x) = 0 has

Let \(f:R \to R\) be such that f is injective and \(f(x)f(y) = f(x + y)\) for \(\forall x,y \in R\). If f(x), f(y), f(z) are in G.P., then x, y, z are in

Let f : X \(\to\) X be such that f [f(x)] = x, for all x\(\in\)X and X\(\subseteq\)R, then

If f(x + 2y, x \(-\) 2y) = xy, then f(x, y) is equal to

The even function of the following is

The period of the function f(x) = cos 4x + tan 3x is

The minimum value of \(f(x) = {e^{({x^4} - {x^3} + {x^2})}}\) is

For what values of x, the function \(f(x) = {x^4} - 4{x^3} + 4{x^2} + 40\) is monotone decreasing?

The domain of the function \(f(x) = \sqrt {{{\cos }^{ - 1}}\left( {{{1 - |x|} \over 2}} \right)}\) is

The domain of definition of the function \(f(x) = \sqrt {1 + {{\log }_e}(1 - x)}\) is

A mapping from IN to IN is defined as follows:
\(f:IN \to IN\)
\(f(n) = {(n + 5)^2},\,n \in IN\)
(IN is the set of natural numbers). Then

Show that sin x is a monotonic increasing function of x in 0 < x < \(\pi\)/2.