If $f(x - y) + f(x + y) = 2f(x)f(y)$ for all $x, y \in R$, then $f(x)$ is .....

If a function $f$ satisfies $f(x+1)+f(x-1)=\sqrt{2} f(x)$,then $f(x+2)+f(x-2)=$

Let $u+v+w=3$, where $u, v, w \in \mathbb{R}$ and $f(x)=u x^2+v x+w$ be such that $f(x+y)=f(x)+f(y)+x y$ for all $x, y \in \mathbb{R}$. Then $f(1)$ is equal to:

Let $f: R \rightarrow R$ be a differentiable function that satisfies the relation $f(x + y) = f(x) + f(y) - 1$ for all $x, y \in R$. If $f'(0) = 2$,then $|f(-2)|$ is equal to:

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 $f: R \rightarrow R$ satisfy $f(x+y)=2^{x} f(y)+4^{y} f(x)$ for all $x, y \in R$. If $f(2)=3$,then $14 \cdot \frac{f^{\prime}(4)}{f^{\prime}(2)}$ is equal to