Let $S=\{1,2,3,4,5,6,7\} .$ Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in S$ and $m . n \in S$ is equal to $......$

If $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{Z}, \mathrm{x}^{2}+3 \mathrm{y}^{2} \leq 8\right\}$ is a relation on the set of integers $\mathrm{Z},$ then the domain of $\mathrm{R}^{-1}$ is 

If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f\left( x \right) = 120,} $ find the value of $n$

Domain of the function $f(x) =$ $\frac{1}{{\sqrt {\ln \,{{\cot }^{ - 1}}x} }}$ is

If function $f(x) = \frac{1}{2} - \tan \left( {\frac{{\pi x}}{2}} \right)$; $( - 1 < x < 1)$ and $g(x) = \sqrt {3 + 4x - 4{x^2}} $, then the domain of $gof$ is

Suppose $f(x) = {(x + 1)^2}$ for $x \ge - 1$. If $g(x)$ is the function whose graph is the reflection of the graph of $f(x)$ with respect to the line $y = x$, then $g(x)$ equals