If $f: R \rightarrow R$ is defined as $f(x)=\frac{3^x+3^{-x}}{2}, \forall x \in R$ and it satisfies $f(x+y)+f(x-y)=a f(x) f(y)$,then $a=$

If a function $f(x)$ satisfies $f(x + y) = f(x) f(y)$ for all $x, y \in N$ such that $f(1) = 3$ and $\sum_{x=1}^n f(x) = 120$,then what is the value of $n$?

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

Let $\sum\limits_{k = 1}^{10} {f(a + k)} = 16(2^{10} - 1),$ where the function $f$ satisfies $f(x + y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2.$ Then the natural number $a$ is

How many functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ are there such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{Z}$?

If $f(x) = x^{2} - 3x + 4$ and $f(x) = f(2x + 1)$,then $x =$