Let $k$ be a field, $f \in k[x_1,...,x_n]$ a irreducible polynomial, define $H = \{(x_1,...,x_{n+1})\mid f(x_1,..,x_n)x_{n+1} = 1\}\subset \Bbb{A}^{n+1}_k$. Prove that:$$k[x_1,...,x_{n+1}]/I(H) \cong k[x_1,...,x_n]_f$$
Where $k[x_1,...,x_n]_f$ means localizing at $f$.
I want to use universal property to prove it. However, I don't have good idea
