Commutativity of Inverse Functions.

Quote:

Originally Posted by ThePerfectHacker

Let, $f(x)$ be a real valued function definited for all reals. Proof (or disproof) that is there exists a function such as $g(f(x))=x$ is true for all reals. Then, $f(g(x))=x$ is always true.

Let $f(x)$ be a sigmoid function something like the cumulative
normal functiom, then $\mathbb{R}$ is mapped one one onto
$(0,1)$ , and so there is an inverse function $g(y)$ from
$(0,1)$ onto $\mathbb{R}$

Then $f(g(x))=x$ holds for all $x \in (0,1)$ but not for all $x \in \mathbb{R}$

That is the domains of $f\circ g$ and $g \circ f$ are not equal.

RonL