Strictly monotone functions such that $x= f(\frac{x^2}{f(x)})$

Question

What are the strictly monotone functions $f\colon (0,\infty)\to (0,\infty)$ which satisfy $x= f(\tfrac{x^2}{f(x)})$ for $x>0$. I cannot find any other than $f(x)=x$.

Answer 1

When $f(x) = a x$, $a>0$, then

$$f \left ( \frac{x^2}{f(x)} \right ) = a \frac{x^2}{f(x)} = a \frac{x^2}{a x} = x $$