I am given the task to prove that f is operation preserving. $f(e^{2πix/n})= [x]_n$ from $G$ $\to$ $\mathbb Z_n$ where $G$ is the group of nth roots of unity under complex multiplication and $\mathbb Z_n$ is the group of integers mod $n$ under modular addition.
I understand that to show a function is operation preserving, I need to show $f(g\cdot h)=f(g)*f(h)$ where "$\cdot$" is the first group's operation and $*$ is the second group's. In my case, "$\cdot$" is complex multiplication and "$*$" is modular addition. The issue I have is that the input of $f$ is not just $x$, but $e^{2πix/n}$, so I am not sure what I'm supposed to replace the $x$ from $[x]_n$ with
At issue here is essentially whether $f$ is well-defined in the first place, that is, is independent of the choice $x$ that gives a particular value $e^{2 \pi i x / n}$; more explicitly, this means that for any $x, y$ such that $e^{2 \pi i x / n} = e^{2 \pi i y / n}$ we need to ensure that $[x]_n = [y]_n$. Once we know that $f$ is well-defined, then checking whether it preserves the operation is just a matter of using the definition of $f$, namely, checking that $$f(e^{2 \pi i x / n} \cdot e^{2 \pi i y / n}) = f(e^{2 \pi i x / n}) +_n f(e^{2 \pi i y / n}) .$$
Remark Another way to think about this is as follows: We're checking that the map $q_n : (\Bbb Z, +) \to (\Bbb Z_n, +_n)$, $a \mapsto [a]_n$ is constant on the fibers of the map $\epsilon: (\Bbb Z, +) \to (G, \,\cdot\,)$, $a \mapsto e^{2 \pi i a / n}$. It is, so there is a map $F : G \to \Bbb Z_n$ satisfying $F \circ \epsilon = q_n$, and unwinding definitions we see that the latter statement is just that the induced map $F$ is just the given map $f$. In fact, if we already know that $q_n$ and $\epsilon$ preserve operations, then we can conclude that $f$ does, too.