What's the official name of "commuting" functions $f:\mathbb{R}\rightarrow \mathbb{R}$, $g:\mathbb{R}^k\rightarrow \mathbb{R}$ with
$$f(g(x_1,\dots x_k)) = g(f(x_1),\dots, f(x_k))$$
For $k=1$ an invertible function $f$ and its inverse $f^{-1}$ with $f(f^{-1}(x)) = f^{-1}(f(x)) = x$ do "commute", but other examples are around:
$f(x) = ax$ and $g(x,y) = x + y$
$f(x) = x^a$ and $g(x,y) = xy$
My main question is: Is it the case, that for a given $g(x,y)$ there is essentially one $f(x)$ that commutes with $g$? If yes: How is it proved? Which requirements have to be posed on $g$ to be able to prove it?
A generalization would be functions $f:\mathbb{R}^k\rightarrow \mathbb{R}$, $g:\mathbb{R}^k\rightarrow \mathbb{R}$ with
$$f(g(x^1_1,\dots x^1_k),\dots , g(x^k_1,\dots x^k_k)) = g(f(x^1_1,\dots x^k_1),\dots , f(x^1_k,\dots x^k_k))$$
for which
$f(x,y) = a(x-y)$ and $g(x,y) = x + y$
$f(x,y) = \frac{x}{y}$ and $g(x,y) = xy$
are an example.