The range of function $f : R \rightarrow  R$, $f(x) = \frac{{{{(x\, + \,1)}^4}}}{{{x^4} + \,1}}$ is

Domain of function $f(x) = {\sin ^{ - 1}}5x$ is

Let $f(x)$ and $g(x)$ be two functions given by $f\left( x \right) = \frac{{2\sin \pi x}}{x}$ and $g\left( x \right) = f\left( {1 - x} \right) + f\left( x \right).$ If $g\left( x \right) = kf(\frac{x}{2})f\left( {\frac{{1 - x}}{2}} \right)$,then the value of $k$ is

The number of one-one function $f :\{ a , b , c , d \} \rightarrow$ $\{0,1,2, \ldots ., 10\}$ such that $2 f(a)-f(b)+3 f(c)+$ $f ( d )=0$ is

The largest interval lying in $\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$ for which the function, $f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {\frac{x}{2} - 1} \right) + \log \left( {\cos x} \right)$  is defined is

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f : A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to $...............$.