1 Many important properties of topological spaces are preserved by continuous maps (but not necessarily open maps): connectedness and compactness come to mind immediately. But more importantly, the most familiar, natural maps that we can define are continuous, but not necessarily open: polynomials $\mathbb{R}^n \to \mathbb{R}^m$.

2 Inverse image of a subgroup under a homomorphism is a subgroup.

3 There is a contravariant functor from the category Set to itself, mapping a set $X$ to its power set $\mathcal{P}(X)$ and sending the morphism $f:X \to Y$ to the inverse image $f^{-1}:\mathcal{P}(Y) \to \mathcal{P}(X)$. A "purely symmetric" function would be a symmetric relation on $X \times Y$, no? Functions are, after all, relations with an extra property that deliberately breaks the symmetry!