We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical systems. Such a combination of under and over-approximations is very valuable for the verification of properties of controlled systems. Over-approximations prove properties correct, while under-approximations can be used for falsification. Coupled, they provide a measure of the conservatism of the analysis. This work introduces a general framework relying on approximations of robust ranges of vector-valued functions, formulated as AE extensions, that can be interpreted as quantified propositions where universal quantifiers (A) precede existential quantifiers (E). This framework allows us to extend for under-approximation many precision refinements that are classically used for over-approximations, such as affine approximations, Taylor models, quadrature formulae and preconditioning methods. We end by evaluating the efficiency and precision of our approach, focusing on the application to the analysis of discrete-time dynamical systems.