Domain of the function $f(x) = {\sin ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\cos ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right)$ is

Range of the function $f(x) = \frac{{{x^2}}}{{{x^2} + 1}}$ is

Let $A=\{(x, y): 2 x+3 y=23, x, y \in N\}$ and $B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $\mathrm{A}$ to $\mathrm{B}$ is equal to ................

If the domain of the function $f(x)=\log _e$ $\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5 \beta-4 \alpha$ is equal to

Suppose that a function $f: R \rightarrow R$ satisfies $f(x+y)=f(x) f(y)$ for all $x, y \in R$ and $f(1)=3 .$ If $\sum \limits_{i=1}^{n} f(i)=363,$ then $n$ is equal to

If $f(x)$ be a polynomial function satisfying $f(x).f (\frac{1}{x}) = f(x) + f (\frac{1}{x})$  and $f(4) = 65$ then value of $f(6)$ is