Let $f \colon X \to Y$ be a morphism of schemes, where $X$ and $Y$ are separated integral Noetherian schemes. Does there necessarily exist a nonempty open affine $U \subset Y$ such that $f^{-1}(U)$ is affine? If $X \to Y$ is not dominant, then the answer is clearly yes (take $U$ to have empty inverse image). Even for dominant morphisms, I feel like this should be related to Chevalley's theorem that the image of a constructible set is constructible, but I can't put it together into a proof.

Note: this question is silly (see comments/answer below). The specific case I had in mind was when $f$ is unramified over the generic point of $Y$; however, if I decide I want the answer to this, I'll ask in a separate question.