$|f(x)g(x)| = |(f(x)||g(x)|$

Question

I was wondering if $|f(x)g(x)| = |f(x)| |(g(x)|$ is true all the time as in the case of real numbers.

I was not convinced enough that that was true.

But I can't think of any counterexample.

Thank you.

Answer 1

Hint: Its is enough to prove it for the squares of the absolute values. And $\lvert z\rvert^2=z\bar z$.

Answer 2

For each $x$, $f(x)$ and $g(x)$ are real (or complex) numbers. So the equality is precisely that of real (or complex) numbers.