Inverse function
Preimages

If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is defined to be the set of all elements of X that map to y:

The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f.

Similarly, if S is any subset of Y, the preimage of S, denoted f − 1 (S) {\displaystyle f^{-1}(S)}, is the set of all elements of X that map to S:

For example, take the function f: R → R; x ↦ x. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

The preimage of a single element y ∈ Y – a singleton set { y } – is sometimes called the fiber of y. When Y is the set of real numbers, it is common to refer to f ({ y }) as a level set.
oulipo:avoid_letter[e], length_constraints:number_paragraphs[3], punctuation:no_comma