Question 165
Determine with proof all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ with the property
$$
f(x) f(y) = 2 f(x + y f(x))
$$
for all positive real numbers $x, y$.
$$
f(x) f(y) = 2 f(x + y f(x))
$$
for all positive real numbers $x, y$.