Let $f\colon X \to Y$ be a morphism of schemes and $J \subseteq \mathcal{O}_Y$ a quasi-coherent ideal. Let $I$ denote the image of $f^* J \to f^* \mathcal{O}_Y = \mathcal{O}_X$. Then $I \subseteq \mathcal{O}_X$ is a quasi-coherent ideal and we have the equality of sets $f^{-1}(V(J)) = V(I)$.
This is easy and should be well-known. I would like to have a reference in the literature, preferred EGA. In EGAI I could only find the affine case, this is Proposition 1.2.2.
PS: Of course this can be also stated as follows: The scheme-theoretic pullback of a closed subscheme has as underlying set precisely the preimage of the underlying closed subset.
EDIT: Here is a deduction from EGA-results: Since $f^* J \to f^* \mathcal{O}_Y \to f^* (\mathcal{O}_Y/J) \to 0$ is exact, we have $\mathcal{O}_X/I = (f^* \mathcal{O}_Y) / I = f^* (\mathcal{O}_Y/J)$. Now apply EGA I, Chapitre 0, 5.2.4.1: $f^{-1}(V(J))=f^{-1}(\mathrm{supp}~ \mathcal{O}_Y/J ) \stackrel{!}{=} \mathrm{supp}~ f^*(\mathcal{O}_Y/J) = \mathrm{supp}~ \mathcal{O}_X/I = V(I)$