The inverse image function associated to $f$ is the function[1]
\[ f^{-1}\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]
defined by[2]
\begin{align*} f^{-1}\webleft (V\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ \text{we have $f\webleft (a\webright )\in V$}\webright\} \end{align*}
for each $V\in \mathcal{P}\webleft (B\webright )$.
