For each of the given maps of coordinate rings, describe the corresponding map of algebraic sets.
$\mathbb{C}[x,y] \to \mathbb{C}[t]$ with $x \to t, y \to t$
$\mathbb{C}[t] \to \mathbb{C}[x,y]$ with $t \to x$
$\mathbb{C}[t] \hookrightarrow \mathbb{C}[t,x,y]/\langle xy -t \rangle$
What does "corresponding map" mean? There's a suggestion "compare to the corresponding maps of Spec", which are defined by $\phi : R \to S \Rightarrow \phi^\# : \text{Spec }S \to \text{Spec }R$ by $\phi^\#(P) = \phi^{-1}(P).$ I know how to compute these, but it doesn't help me reverse engineer what the algebraic sets map might be.
Update: I believe for $\phi : R \to S,$ the correspondence is either $V(I) \to V(\phi(I))$ or contravariantly, $V(I) \to V(\phi^{-1}(I)).$ Which one is better, and could there be a different interpretation?
As you may know, given algebraic sets $X \subseteq \mathbb{A}^m$, $Y \subseteq \mathbb{A}^n$, the natural map $$\mathrm{Mor}(X,Y) \longrightarrow \mathrm{Hom}_k(k[Y],k[X])$$ is a bijection (here $k[X]$ and $k[Y]$ are the coordinate rings of $X$ and $Y$).
Note that $\mathbb{C}[x,y]$, $\mathbb{C}[t]$ and $\mathbb{C}[x,y]/\langle xy-t \rangle$ are the coordinate rings of $\mathbb{A}^2$, $\mathbb{A}^1$ and $V(xy-t) \subset \mathbb{A}^3$ respectively, so they are asking you to give the inverse images of those $\mathbb{C}$-algebra homomorphisms under the above bijection.
Respectively, the morphisms are
$$\mathbb{A}^1 \to \mathbb{A}^2; \quad t \mapsto (t,t),$$ $$\mathbb{A}^2 \to \mathbb{A}^1; \quad (x,y) \mapsto x,$$ $$V(xy-t) \to \mathbb{A}^1; \quad (t,x,y) \mapsto t.$$