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) = $

If $f: R-\{0\} \rightarrow R$ is defined by $3 f(x)+4 f\left(\frac{1}{x}\right)=\frac{2-x}{x}$, then find the value of $f(3)$.

If $f(x) = \frac{x}{x - 1} = \frac{1}{y}$,then $f(y) = $

$A$ function $f: R \rightarrow R$ is such that $f(1)=2$ and $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed in terms of $f(1)$, $f(2)$, and $f(4)$ is

The number of functions $f : \{1, 2, 3, 4\} \rightarrow \{ a \in \mathbb{Z} : |a| \leq 8 \}$ satisfying $f(n) + \frac{1}{n} f(n+1) = 1$ for all $n \in \{1, 2, 3\}$ is

Let $f: R-\{0\} \rightarrow R$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y$ where $f(y) \neq 0$. If $f^{\prime}(1)=2024$,then: