Question 149
Let $\mathbb{R}_{\gt 0} = \{x \in \mathbb{R} \mid x \gt 0\}$ denote the set of positive real numbers.
Find all pairs $f, g: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ of functions satisfying
$$
f(g(x)) = f(x)g(x), \quad f(x) = x(1 + g(x))
$$
and such that the sequence $g(x), g(g(x)), g(g(g(x))), \dots$ takes finitely many different values for all $x \in \mathbb{R}_{\gt 0}$.
Find all pairs $f, g: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ of functions satisfying
$$
f(g(x)) = f(x)g(x), \quad f(x) = x(1 + g(x))
$$
and such that the sequence $g(x), g(g(x)), g(g(g(x))), \dots$ takes finitely many different values for all $x \in \mathbb{R}_{\gt 0}$.