Mapping/x^2-1/Image and preimage/No linear subspaces/Exercise

We consider the mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}-1.$

Show that for this mapping, neither the image of a linear subspace ${}U\subseteq \mathbb {R}$ must be a linear subspace, nor the preimage of a linear subspace must be a linear subspace.

Create a solution