I have a morphism $f:A\longrightarrow B$ of finitely generated $\mathbb{C}$-algebras. I have proven, using Zariski's lemma, that the inverse image of a maximal ideal $M \subset B$ is a maximal ideal $f^{-1}(M) \subset A$, hence the induced map $\operatorname{Spec}(f) : \operatorname{MaxSpec}(B) \longrightarrow \operatorname{MaxSpec}(A)$ defined by $\operatorname{Spec}(f)(M) = f^{-1}(M)$ is well-defined, and even continuous for the Zarkiski topology.
Now I need to prove that
$f$ is surjective $\Longleftrightarrow$ $\operatorname{Spec}(f)$ is injective
I have proven the implication "$\Longrightarrow$", since I think it's only a matter about set theory. I am stuck in the other implication. By factoring $f : A \longrightarrow A/\ker(f) \longrightarrow B$, I think I may assume that $f$ is injective, but I don't know how to proceed with that.