I want to prove that $\mathbb{A}^1$ is not isomorphic to $X =\mathbb{A}^1 \setminus \{p\}$ where $p \in \mathbb{A}^1$.
Assuming I can take $p = 0$ by change of coordinates, at first I tried proving that $X$ is not an affine variety, but I found in another post (https://math.stackexchange.com/q/880955) that it actually is.
So now I think I should use the function rings but I'm not sure how to proceed...
Hint: Right, show that $k[x]$ and $k[x,x^{-1}]$ are not isomorphic. You can do this by showing that the latter ring has no single element that generates it (as a ring). I think it is helpful to keep track of min and max degrees of the terms of an element of $k[x, 1/x]$. You can first show that $x + 1/x$ doesn't generate $k[x,1/x]$, the figure out why this didn't work and generalize.
Alternative approach (cleaner): You can show that the only ring maps from $k[x,1/x] \to k[x]$ have to send $x$ and $1/x$ to units, and thus these maps can't be surjective.