Suppose that $f:X\rightarrow Y$ is a morphism between two affine varieties over an algebraically closed field $K$.
I believe that if the corresponding morphism of $K-$algebras, $f^\ast:K[Y]\rightarrow K[X]$ is injective, it is not necessarily true that $f:X\rightarrow Y$ must be surjective but I have yet to come up with a counterexample.
Is there such a counterexample?