What is the general solution to the equations f(g(x))=h(x) and g(f(x))=h(x), or how do you find the particular soltuion to the equations given a function f (given any f what is the general form of h(x) such that g(x) exists)? My thoughts on the topic: Specific example: Suppose we have f(x)=x+1 and h(x)=x. Then g(x)=x-1. Suppose instead h(x)=x+2. Then, f(g(x))=g(x)+1=x+2 g(x)=x+1 g(f(x))=x+1+1=x+2 so it checks. Now suppose h(x)=3x f(g(x))=g(x)+1=3x g(x)=3x-1 g(f(x))=3(x+1)-1=3x+2 doesn't check. More general case: Suppose now f(x)=x2 f(g(x))=[g(x)]2=h(x) g(x)=sqrt(h(x)) g(f(x))=[squ]h(x2)=h(x) So g(x) exists for a given h(x) if h(x2)=[h(x)]2 I don't think I can solve this any further. I feel resigned to the fact that I must check to see that this holds for some specific h(x) rather than finding the general solution. Certainly, h(x)=xk works. Any further thoughts? Edit: Oh by the way, h(x)=x of course works for all f provided that f has an inverse. So any general form should be able to reduce to h(x)=x.