Consider the map of affine schemes $ \operatorname{Spec}\left( \dfrac{\mathbb{C}[x,y]}{\langle xy\rangle }\right)\stackrel{f}{\rightarrow} \operatorname{Spec}\mathbb{C}[t] $ whose corresponding map of rings is $ \mathbb{C}[t]\stackrel{f^*}{\rightarrow} \dfrac{\mathbb{C}[x,y]}{\langle xy\rangle } \hspace{10 mm}(\star) $ where $t\mapsto x+y$.
It is clear geometrically that $f^{-1}(t-c)$ is two points for $c\not=0$ while for $c=0$, it is the double point $\mathbb{C}[x]/\langle x^2\rangle$.
But how does one deduce this only from the map of rings ($\star$)?
Thank you.