Let $X:=Z(xz-y^2)\subset\mathbb{P}^1$ and the following morphism:
\begin{align*} \phi:X &\to\mathbb{P}^1\\ (a:b:c) &\mapsto (a:b)\,\,\text{for } (a,b)\neq (0, 0)\\ (a:b:c) &\mapsto (b:c)\,\,\text{for } (b,c)\neq (0,0) \end{align*}
I'm trying to check whether or not $\phi$ is a finite morphism (i.e., if $\phi^*$ is finite morphism of $k$-algebras).
My usual approach is this: every finite morphism is surjective and has finite fibers, so if any of these two properties fail for $\phi$, then $\phi$ it can't be finite.
This strategy doesn't work here, so I'm stuck. Any ideas?