If $f(x)=\frac{2^{x}+2^{-x}}{2}$,then $f(x+y) \cdot f(x-y)$ is

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

Let $f, g$ and $h$ be the real-valued functions defined on $\mathbb{R}$ as $f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{cases}$,$g(x) = \begin{cases} \frac{\sin(x+1)}{x+1}, & x \neq -1 \\ 1, & x=-1 \end{cases}$ and $h(x) = 2[x] - f(x)$,where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is

If $f(x) = \cos (\log x)$,then $f(x^2)f(y^2) - \frac{1}{2}\left[ f\left( \frac{x^2}{y^2} \right) + f(x^2y^2) \right]$ has the value

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be a function such that $f(x) > 0$ for all $x \in R$,and $f(x+y)=f(x) f(y)$ for all $x, y \in R$. Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31})=64 f(a_{25})$,and $\sum_{i=1}^{50} f(a_i)=3(2^{25}+1)$,then the value of $\sum_{i=6}^{30} f(a_i)$ is:

If $X$ and $Y$ are two non-empty sets where $f: X \to Y$ is a function defined such that $f(C) = \{f(x) : x \in C\}$ for $C \subseteq X$ and $f^{-1}(D) = \{x : f(x) \in D\}$ for $D \subseteq Y$,then for any $A \subseteq X$ and $B \subseteq Y$,which of the following is true?