The range of the function $f(x){ = ^{7 – x}}{\kern 1pt} {P_{x – 3}}$ is

Let $\mathrm{f}: N \rightarrow N$ be a function such that $\mathrm{f}(\mathrm{m}+\mathrm{n})=\mathrm{f}(\mathrm{m})+\mathrm{f}(\mathrm{n})$ for every $\mathrm{m}, \mathrm{n} \in N$. If $\mathrm{f}(6)=18$ then $\mathrm{f}(2) \cdot \mathrm{f}(3)$ is equal to :

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 $……………$.

A real valued function $f(x)$ satisfies the function equation $f(x – y) = f(x)f(y) – f(a – x)f(a + y)$ where a is a given constant and $f(0) = 1$, $f(2a – x)$ is equal to

Numerical value of the expression $\left| {\;\frac{{3{x^3} + 1}}{{2{x^2} + 2}}\;} \right|$ for $x = – 3$ is