Question 126
Let $\mathbb{R}_{\gt 0} = \{x \in \mathbb{R} \mid x \gt 0\}$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ satisfying
$$
f(x)f(y + f(x)) = f(1 + xy)
$$
for all $x, y \in \mathbb{R}_{\gt 0}$.
(Otgonbayar Uuye)
$$
f(x)f(y + f(x)) = f(1 + xy)
$$
for all $x, y \in \mathbb{R}_{\gt 0}$.
(Otgonbayar Uuye)