Let $K$ be a field and let $F=\frac{P}{Q}\in K(X)$ be a rational fraction, for simplicity we denote also by $F$ the rational function associated to the rational fraction $F$. It is clear that if $P$ and $Q$ are both even or both odd polynomial functions, then $F$ is an even rational function and we have also that if one of the two polynomial functions is even and the other is odd, then $F$ must be odd. Now is the converse true, I mean do we have that if $F$ is an even rational function then necessarely both $P$ and $Q$ are both odd or both even and that if $F$ is odd then necesserely one is odd and the other is even ?

I think it is true, but i don't know how to prove it, suppose that $F$ is even then

$\frac{P(x)}{Q(x)}=\frac{P(-x)}{Q(-x)}$

Hence

$P(x)Q(-x)=P(-x)Q(x)$

But how to go from here?

I think it is true, but i don't know how to prove it, suppose that $F$ is even then

$\frac{P(x)}{Q(x)}=\frac{P(-x)}{Q(-x)}$

Hence

$P(x)Q(-x)=P(-x)Q(x)$

But how to go from here?