Are there real-valued functions $f$ and $g$ which are neither each other's inverses, the identity, nor linear, yet exhibit the behaviour $$f\circ g = g \circ f?$$
Examples such as $f(x) = 2x$ and $g(x)=3x$ are "trivial" in this sense.
Moreover, given a function $f$, can one go about obtaining an example of a function $g$ which commutes with $f$?
I suppose this would be similar to fixed point iteration? E.g. if $f\colon\mathbb R\smallsetminus\{1\}\to\mathbb R$ is defined by $f(x) = 2x/(1-x)$, I would need a function $g$ such that $$g(x) = f^{-1}\circ g\circ f =\frac{g(\frac{2x}{1-x})}{2+g(\frac{2x}{1-x})},$$ so maybe choosing an appropriate "starting function" $g_0$ and finding a fixed point of $g_{n+1} \mapsto f^{-1}\circ g_n\circ f$ would be a possible strategy, but I can't seem to find a suitable $g_0$.
