Let $f$ be a function defined by $f(xy) = \frac{f(x)}{y}$ for all positive real numbers $x$ and $y$. If $f(30) = 20$,then $f(40) = $

Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in R$,and $g: R \rightarrow(0, \infty)$ be a function such that $g(x+y)=g(x) g(y)$ for all $x, y \in R$. If $f\left(\frac{-3}{5}\right)=12$ and $g\left(\frac{-1}{3}\right)=2$,then the value of $\left(f\left(\frac{1}{4}\right)+g(-2)-8\right) g(0)$ is.

Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2x+1}{3}$. If $\alpha$ is an element in the domain of $f$ whose image is $\frac{1}{\alpha}$,then the sum of all possible values of such $\alpha$ is

Let $f: R \rightarrow R$ be such that $f$ is injective and $f(x)f(y) = f(x+y)$ for all $x, y \in R$. If $f(x), f(y),$ and $f(z)$ are in $GP$, then $x, y,$ and $z$ are in:

Given the function $f(x) = \frac{a^x + a^{-x}}{2}, (a > 2)$,then $f(x + y) + f(x - y)$ is equal to

If $f : \mathbb{Z} \rightarrow \mathbb{Z}$ is defined by $f(x) = x^{9} - 11 x^{8} - 2 x^{7} + 22 x^{6} + x^{4} - 12 x^{3} + 11 x^{2} + x - 3, \forall x \in \mathbb{Z}$,then $f(11) = $