For any two functions $f(x)$ and $g(x)$ we are given $f \circ g = g \circ f$. What does this imply?
I found that $f(x) = g(x)$, $f(x) = g^{-1}(x)$ and $ f(x) = x \ (\neq g(x))$ are some of the solutions. However, are they the only functions satisfying this? If so, how can we prove it?
Clarification : $f \circ g$ denotes the composition of $f$ and $g$, i.e, $f(g(x))$
They are not the only functions satisfying that, for example, $f(x)=2x, g(x)=3x$ satisfies it too.
There are lots of other examples. In particular, suppose
$f(x) = x$ when $x \ge 0$ and $f(x) < 0 $ when $x<0$
$g(x) = x$ when $x \le 0$ and $g(x) > 0 $ when $x>0$
The idea here is that each of $f$ and $g$ is the identity function on a part of the domain that's mapped to itself by the other.
