If both $f(x)$ and $g(x)$ are one to one functions and $g(f(x))= f(g(x)) = x$, why does this prove they are inverse of each other?

Question

Both $f(x)$ and $g(x)$ are one to one functions and $g(f(x)) = f(g(x)) = x$ why does this prove they are inverse of each other? I understand function composition but the way they overlap confuses the hell out of me. Can anyone who answers the question also guide me into being able to read these function compositions without getting confused with the overlapping?

Answer 1

That's what the term "inverse function" means. You put $3$ into $f$ and get $95$; you put $95$ into the inverse of $f$ and get $3.$ And so on.

Answer 2

Simply because, by definition: $$f^{-1}(f(x))=x$$ and $$f(f^{-1}(x))=x$$

For example, if $f(x)=y$, then $f^{-1}(y)=x$