Suppose $\quad f : R \rightarrow(0, \infty)$ be a differentiable function such that $5 f ( x + y )= f ( x ) \cdot f ( y ), \forall x , y \in R$. If $f(3)=320$, then $\sum \limits_{n=0}^5 f(n)$ is equal to :

The period of the function $f(x) = e^{x -[x]+|cos\, \pi x|+|cos\, 2\pi x|+....+|cos\, n\pi x|}$ (where $[.]$ denotes greatest integer function); is:-

Consider the identity function $I _{ N }: N \rightarrow N$ defined as $I _{ N }$ $(x)=x$  $\forall $  $x \in N$ Show that although $I _{ N }$ is onto but $I _{ N }+ I _{ N }:$  $ N \rightarrow N$ defined as $\left(I_{N}+I_{N}\right)(x)=$ $I_{N}(x)+I_{N}(x)$ $=x+x=2 x$ is not onto.

The mid-point of the domain of the function $f(x)=\sqrt{4-\sqrt{2 x+5}}$ real $x$ is

Which pair $(s)$ of function $(s)$ is/are equal ?

where $\{x\}$ and $[x]$ denotes the fractional part $\&$ integral part functions.

If $\,\,f(x) = \left\{ {\begin{array}{*{20}{c}}
  {3 + x;\,\,\,\,\,x \geqslant 0} \\ 
  {2 - 3x;\,\,\,\,\,x < 0} 
\end{array}} \right.$ then $\mathop {\lim }\limits_{x \to 0} f(f(x))$ is equal to -