TITLE: Is there a notation for the inverse image function itself?
QUESTION [5 upvotes]: Let $f$ be a function.
If $B \subset im(f)$ then the inverse image of $B$ under $f$ in the current standard notation is
$f^{-1}(B) = \{x \in dom(f) : f(x) \in B\}$.
But $f^{-1}$ does not denote the inverse image function !
Using a kind of the typed lambda notation the inverse image function
-- the name $preim$ (from preimage) may be used - can be defined as follows:
$preim := F[B \in P(im(f)), \{x \in dom(f): f(x) \in B \}]$.
But in the common case (if $f$ is not necessary bijective):
$f^{-1} = F[y \in im(f)), \{x \in dom(f): f(x) = y \}]$.

REPLY [2 votes]: Some common notations for the converse of a relation $R$:
$R^\text{T}$, $\breve{R}$, $\check{R}$.
Now these aren't quite what you're asking for. As you note, if $f:X\rightarrow Y$, then the function mapping $P(X)\rightarrow P(Y)$ (powersets) is different. However, another standard notation for $f(A)$, with $A\subseteq X$, is $f``A$. So you might write
$$\breve{f}``$$
for your function. I admit I've never seen this particular combination in print.