BALTIC-WAY 1993 Problem 6

Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ such that $2<x<4$ and satisfy $2<f(x)<4$, $2<g(x)<4, f(g(x))=g(f(x))=x$ and $f(x) \cdot g(x)=x^{2}$ for all such values of $x$. Prove that $f(3)=g(3)$.