If $a+\alpha=1, b+\beta=2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}$ for $x \neq 0$,then the value of the expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ 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?

If $f: R-\{0\} \rightarrow R$ is defined by $f(x)=x+\frac{1}{x}$ and if $f^k(x)=[f(x)]^k$ for $k \geq 1$,then find the value of $f^4(x)-f(x^4)-4f^2(x)$.

If $f(x) = \sin \log x$,then the value of $f(xy) + f\left( \frac{x}{y} \right) - 2f(x) \cos \log y$ is equal to

If $R \subset A \times B$ and $S \subset B \times C$ be two relations,then $(S \circ R)^{-1} = $

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression with mean zero,then the value of $f(g(\frac{1}{n} \sum_{i=1}^{n} f(a_i)))$ is equal to.