Consider a polynomial $f(x)$ with real coefficients having the property $f(g(x)) = g(f(x))$ for every polynomial $g(x)$ with real coefficients. Determine and prove the nature of $f(x)$.
My guess is that $f(x)$ is its own inverse. Could anyone shed some more fire on this for me?
Also, what if $f(x)$ is not its own inverse?