No method for the computation of a reliable subset of the range of vector-valued functions is available today. A method for computing such inner approximations is proposed in the specific case where both domain and co-domain have the same dimension. A general suffi- cient condition for the inclusion of a box inside the image of a box by a continuously differentiable vector-valued is first provided. This sufficient condition is specialized to a more efficient one, which is used in a specific bisection algorithm that computes reliable inner and outer approximations of the image of a domain defined by constraints. Some experimentations are presented.