If $f(x + y) = f(x)f(y)$ for all $x$ and $y$ and $f(5) = 2$,$f'(0) = 3$,then $f'(5)$ will be

If $f(x_1) - f(x_2) = f\left( \frac{x_1 - x_2}{1 - x_1 x_2} \right)$ for $x_1, x_2 \in [-1, 1]$,then $f(x)$ is

If $f(x)$ and $g(x)$ are two real-valued functions such that $f(g(x+y)) = f(g(x)) + f(g(y))$,$g(1) = 2$,and $f(2) = 1$,then the function $g(f(x))$ is discontinuous on the set

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 $\sum_{i=1}^{n} f(i)=363$,then $n$ is equal to

Suppose $f$ is a function satisfying $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{N}$ and $f(1) = \frac{1}{5}$. If $\sum_{n=1}^m \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$,then $m$ is equal to $...............$.

Let $f$ be a polynomial function such that $\log_2(f(x)) = (\log_2 (2 + \frac{2}{3} + \frac{2}{9} + \dots \infty)) \cdot \log_3 (1 + \frac{f(x)}{f(1/x)}), x > 0$ and $f(6) = 37$. Then $\sum_{n=1}^{10} f(n)$ is equal to: