Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ with $2<x<4$ and satisfy: $2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2$ for all $2<x<4$. Prove that $f(3)=g(3)$.
Source: Baltic Way 1993
Tags: function, algebra proposed, algebra
Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ with $2<x<4$ and satisfy: $2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2$ for all $2<x<4$. Prove that $f(3)=g(3)$.