True or false? If $f$ and $g$ are entire functions such that $f(z) g(z) =1$ for all $z$, then $f$ and $g$ are constants

Question

This question was asked in complex analysis quiz and I was unable to solve it at that time , so I am asking it here for help.

State true/false with proper explanations: If $f$ and $g$ are entire functions such that $f(z) g(z)=1$ for all $z$ then $f$ and $g$ are constants.

Unfortunately, I am clueless on how to approach this question and would not not be able to provide an attempt, except: I tried to use Picard's theorem but realized it could not be used, as an entire function can omit one value.

Kindly help.

Answer 1

$f(z)=e^{z}, g(z)=e^{-z}$ shows that $f$ and $g$ need not be constants.