Take scheme morphism $f: X\to Y$ and suppose $f$ surjective. If $y \in Y$ can one find affine open $V \subset Y$ containing $y$ and affine open $U \subset X$ such $f(U) = V$ ? Thank you.

Later: Very good answer of Kevin shows it is not true. Is there hypothese which make it true ? For example $X$ irreducible and/or $f$ faithfuly flat ?