If $f : R \to R$ is such that $f(x + y) = f(x) + f(y)$ for all $x, y \in R$,$f(1) = 7$ and $\sum_{r=1}^{n} f(r) = 14112$,then $n$ is equal to:

$A$ function $f: R \rightarrow R$ satisfies $f\left(\frac{x+y}{3}\right) = \frac{f(x)+f(y)+f(0)}{3}$ for all $x, y \in R$. If the function $f$ is differentiable at $x=0$, then $f$ is:

Given the function $f(x) = \frac{a^x + a^{-x}}{2}$,where $a > 2$. Then $f(x + y) + f(x - y) = $

Let $f(x)$ be defined for all $x > 0$ and be continuous. Let $f(x)$ satisfy $f\left( \frac{x}{y} \right) = f(x) - f(y)$ for all $x, y > 0$ and $f(e) = 1$. Then:

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

Let $f: N \times N \rightarrow N$ be a function such that $f(1,1)=2$,$f(m+1, n)=f(m, n)+2(m+n)$,and $f(m, n+1)=f(m, n)+2(m+n-1)$ for all $m, n \in N$. Find the value of $f(2,2)$.