Confusion about what exactly a rational function on an integral scheme does

Question

Let $X$ be an integral scheme of finite type over an arbitrary field $k$. I am interpreting rational functions on $X$ as elements of the stalk $\mathcal{O}_{X, \eta}$ at the generic point $\eta$ of $X$. Does this rational function somehow determine a morphism from $X$ to $\mathbb{P}_{k}^{1}$? If so, how?