Why is this morphism a homeomorphism?

Question

Why is the morphism of affine varieties $$ \varphi : \mathbb{A}^1 \to \mathcal{V}(X^3-Y^2) \subset \mathbb{A}^2, \quad r \mapsto (r^2,r^3),\;$$ a homeomorphism? It is obviously bijective, so it remains to show that the image of an arbritary variety $V \subset \mathbb{A}^1$ under $\varphi$ is a subvariety of $\mathcal{V}(X^3-Y^2)$.