Composing trigonometric functions

Question

Let $f(x)=\sin(x)$. If $g$ and $h$ are functions on $\mathbb{R}$ such that $g(f(x))= h(f(x))$, can we conclude $g=h$ ?

Can we actually compare $g$ and $h$?

I am confused. Please, help me.

Answer 1

The range of $sin(x)$ is only [-1,1], not all of $\mathbb R$ . It is possible to come up with functions $f$ and $g$ that coincide on [-1,1] but not on other values of $\mathbb R$, so I would think the answer is no. Now if $f$ and $g$ were continuous, that might not be the case, but the problem statement did not specify that.

Answer 2

In general no! $-1\le f(x)\le 1$ for all $x$. For $|x|\gt 1$, we have no information about $g$ or $h$, so they could be completely different. However if $g$ and $h$ are analytic, then they would be equal.