Suppose that we are given a polynomial
$$ f(x) \colon= a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n, $$ where $a_i \in {\Bbb R}~ {\mathrm{or}}~{\Bbb C}$. Then it would be asked that
Q. What is the characterisation (possibly understandable or reasonable) of the function $g(x)$ such that $f(g(x)) = g(f(x))$ holds?
Originally I stumbled into the question posed on a certain website, which asks the relations between $f(x)$ and $g(x)$ when $f(g(x)) = g(f(x))$ holds.
Then, I got an idea of determining $g(x)$ once $f(x)$ is given with a polynomial. This will help proceed with the generalisation, for Weierstrass theorem ensures the uniform approximation for an arbitarary continuous functions by polynomials.