# Twisted section corresponding to rational function

I am following Vakil's notes on Foundations of algebraic geometry. We have just introduced twisting sheaves $${\cal O}(n)$$ on $$\mathbb P^1_k$$ by gluing the structure sheaves on standard open subsets with a non-standard map.

Later in the text, we take a rational function $$s \in {\cal O}_X(U)$$ and associate to it a rational section of $${\cal O}(n)$$, but the procedure on how this is done is not explained. It is not clear to me how to get a section of $$s$$, since we constructed $${\cal O}(n)$$ by gluing and there seems to be some choice related to which chart of $${\cal O}(n)$$ one chooses to interpret the function $$s$$. As $${\cal O}(n)$$ is non-trivial, this choice will produce different rational functions.

Example: How can we interpret $$1/x_0$$ as a section of $${\cal O}(1)$$? Or of $${\cal O}(-1)$$? We can think of $$1/x_0$$ as defined on an open subset of $$D(x_0)$$ as well as an open subset of $$D(x_1)$$. If we start with $$D(x_0)$$, then it gives a regular function $$1/x_0$$ there, and this function corresponds to $$x_1^2$$ on $$D(x_1)$$ if we are working with $${\cal O}(1)$$, and to $$1$$ on $$D(x_1)$$ if we are working with $${\cal O}(-1)$$.