I'm working on some old papers for a math Olympiad, and I ran into a problem I cannot figure out Let $f(x) =x+ 2\pi$, find a non-polynomial function that commutes with f(x) in composition.
I've tried all the obvious functions, trig, inverse trig, exponential, logarithmic, some simple rational functions, and in each case it either lead to something dependent on x, hence it wouldn't commute, or the conditions to commute would turn it into a polynomial. For example: Let $g(x)=\frac{ax+b}{cx+d}$ leads to a=d, c=0, i.e. $g(x)=x+b'$
Then the question goes on to ask to show that there are infinitely many functions that commute with f(x).
You want a function $g$ such that $g(x+2\pi)=g(x)+2\pi$ for all $x$. So define $g$ any way you like on the interval $[0,2\pi)$, and then repeatedly apply $g(x+2\pi)=g(x)+2\pi$ to define $g$ on the rest of $\mathbb R$. For example, choosing $g$ to be identically $0$ on $[0,2\pi)$, you'll get $g(x)=2\pi\lfloor x/2\pi\rfloor$ for all $x$.