Prove that if $P ∈ C(\overline{K})$ is a pole of $f ∈ K(C)$, then $v_P(f) < 0$ and $P$ is a zero of $1/f$.

Question

Let $C$ is a genus-2 hyperelliptic curve defined over a finite field $K$.

As $P$ is pole of $f$, $f \in K(C)$ such that $f\equiv \frac{f_1}{f_2}$, then $f_2(P)=0$, so $\left(\frac{1}{f}\right)(P)=\frac{f_2(P)}{f_1(P)}=0$, this is enough?

Because I do not know that $f_1 \in \mathcal{O}_{P,\overline{K}}(C)$, or can I say that we can choose the representative of the class of $f$ such that $f_1 \in \mathcal{O}_{P,\overline{K}}$.

And for the other part, I have some theorems for example $v_p(f)=max \{m\in \mathbb{N}\;| \;\frac{f}{t_p^{m}} \in \mathcal{O}_{p,\overline{K}(C)} \}$, $t_p \in \mathcal{O_{P,\overline{K}}}$ be the uniformiset.

But I'm not sure how to go on. Any hints?