Properties on operators / functions

Question

If $f(g(x)) = g(f(x))$ whats the name of the properties on $f$ and $g$

Answer 1

You can say that $f$ and $g$ commute. "Commuting" makes sense with respect to a given operation. In your case, it is the operation $\circ$ of composition, and $f(g(x)) = g(f(x))$ for all $x$ is also written just as $f\circ g = g \circ f$. Writing $f(x,y) =f(y,x)$ means that $f$ is symmetric, which is something else.