Let $k$ be a field and consider the degree $n$ map $f: \mathbb P^1_k \to \mathbb P^1_k$ given by $(x:y) \mapsto (x^n: y^n)$. I want to show that $f^*(\mathcal O_{\mathbb P^1_k}(1)) \simeq \mathcal O_{\mathbb P^1_k}(n)$.
My attempt is as follows: by definition, we have to calculate $$f^*(\mathcal O_{\mathbb P^1_k}(1)) = f^{-1}(\mathcal O_{\mathbb P^1_k}(1)) \otimes_{f^{-1}(\mathcal O_{\mathbb P^1_k})} \mathcal O_{\mathbb P^1_k},$$ so we have to describe the sections of this sheaf over every affine open set $U = {\rm Spec}(R)$ and show that they agree with $\mathcal O_{\mathbb P^1_k}(n)({\rm Spec}(R))$. How to describe these sections? I guess the sections of $f^{-1}(\mathcal O_{\mathbb P^1_k}(1))({\rm Spec}(R)))$ given by rational functions of the form $g(x^n,y^n)/h(x^n,y^n) \in R(x,y)$ where $g(x,y)$ and $h(x,y)$ are homogeneous polynomials with $deg(g) - deg(h) = 1$; is this correct? How to get this? If this is true, then I guess we can compute the tensor product in the display above to get that $f^*(\mathcal O_{\mathbb P^1_k}(1))({\rm Spec}(R))$ consists of rational functions of the form $$\frac{g(x^n,y^n)}{h(x^n,y^n)} \cdot \frac{p(x,y)}{q(x,y)},$$ where $p$ and $q$ are homogeneous of the same degree. But why are these all the rational functions $\frac{r(x,y)}{s(x,y)}$, where $r$ and $s$ are homogeneous with $deg(r)-deg(s)=n$?