The notion of $f^{-1}$, so far as I have seen it treated, seems to only be defined for intervals where $f$ is one-to-one. I imagine there's a more general concept for the function that describes the set of all possible origin points.
I found the term "full inverse" online but only the site that used it itself showed up when searching for other references using it in the same context. I wonder if there's another more common name for such a function?
$f^{-1}(A)=\{x \mid f(x) \in A\}$ is called the inverse image or preimage of $A$ under $f$.