In this article we will investigate nonmeasurability with respect to some σ-ideals of images of subsets of a Polish space X via selected mappings from X to another Polish space Y.In particular, we answer the following question: “Is it true that there exists a subset of the unit disc in the real plane such that there are continuum many projections onto lines with Lebesgue measurable images and continuum many projections without this property?”. It is known that there exists continuous function f:[0,1]→[0,1] such that for every Bernstein set B⊆[0,1] we have f[B]=[0,1]. We show the relative consistency with ZFC of the fact that the above result is not true for some N- or M-completely nonmeasurable sets, even if we take less than c many continuous functions.