$\mathbb{C}(X)$ as a finite extension of $\mathbb{C}(Y)$

Question

Let $X$ and $Y$ affine varieties. If $\mathbb{C}(X)$ (the field of rational functions on $X$) is a finite extension of $\mathbb{C}(Y)$, we can assume that the extension is generated by a single $f\in \mathbb{C}(X)$. However, I read in a proof that we can also assume that $f \in O(X)$, where $O(X)$ is the coordinate ring of $X$. Why is this true?