The inverse image function associated to $f$ is the function1
\[ f^{-1}\colon \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\webright ) \]
defined by2
\begin{align*} f^{-1}\webleft (V\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \text{we have $f\webleft (x\webright )\in V$}\webright\} \end{align*}
for each $V\in \mathcal{P}\webleft (Y\webright )$.
1Further Notation: Also written $f^{*}\colon \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\webright )$.
2Further Terminology: The set $f^{-1}\webleft (V\webright )$ is called the inverse image of $V$ by $f$.
