Hartshorne say that a $ \phi : V \to W$ is a morphism of variety if it is continuous and for every regular function $f$ on $W$, the map $ f \circ \phi^{-1} $ is also regular function.
Using this definition, how do we show that $\phi : Z(XY-1) \to \mathbb{A}^1 -\{0\}$ s.t.$ (x,\frac{1}{x}) \mapsto x$ and the inverse map $ \psi : \mathbb{A}^1 -\{0\} \to Z(XY-1)$ s.t. $ x \mapsto (x,\frac{1}{x}) $ are both morphisms of varieties.