Question 138
Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following conditions:
i) $f(2x) \ge 2f(x)$ for all $x \gt 0$;
ii) $f$ is strictly increasing $(0; +\infty)$.
iii) $f(f(x)f(y) + x) = f(xf(y)) + f(x)$ for all $x, y \gt 0$.
i) $f(2x) \ge 2f(x)$ for all $x \gt 0$;
ii) $f$ is strictly increasing $(0; +\infty)$.
iii) $f(f(x)f(y) + x) = f(xf(y)) + f(x)$ for all $x, y \gt 0$.