Question 101
Find all functions $f: \mathbb{R}_{ \gt 0} \rightarrow \mathbb{R}$ such that
$$
f\left(\frac{x}{y}\right)=f(x)+f(y)-f(x) f(y)
$$
for all $x, y \in \mathbb{R}_{ \gt 0}$. Here, $\mathbb{R}_{ \gt 0}$ denotes the set of all positive real numbers.
$$
f\left(\frac{x}{y}\right)=f(x)+f(y)-f(x) f(y)
$$
for all $x, y \in \mathbb{R}_{ \gt 0}$. Here, $\mathbb{R}_{ \gt 0}$ denotes the set of all positive real numbers.