My current exercise is to determine whether the set $\{(x, \frac{1}{x}) \mid x \in \mathbb{R} \setminus \{0\}\} \subseteq \mathbb{R}^2$ is affine algebraic.
My guess is that it is not, but I currently do not know how to prove it. In class, we learned some basic facts about the Zariski topology and Hilbert's Nullstellensatz.
The standard approaches I could think of (showing the set is not closed in the standard topology and restricting the polynomial to one coordinate) seem to fail, so I currently lack a general approach to tackle this approach.
It is actually algebraic, since it is the set where the polynomial $XY-1$ vanishes.