Preimage

The preimage of an element ⁠${\displaystyle y}$⁠ of the codomain of ⁠${\displaystyle f}$⁠ is the set of all elements ⁠${\displaystyle x}$⁠ of the domain of ⁠${\displaystyle f}$⁠ such that ⁠${\displaystyle f(x)=y}$⁠; it is empty if ⁠${\displaystyle y}$⁠ does not belong to the image of ⁠${\displaystyle f}$⁠. The preimage of a subset ⁠${\displaystyle T}$⁠ of the codomain of ⁠${\displaystyle f}$⁠ is the set of all elements ⁠${\displaystyle x}$⁠ of the domain of ⁠${\displaystyle f}$⁠ such that ⁠${\displaystyle f(x)\in T}$⁠. The preimage of the codomain of ⁠${\displaystyle f}$⁠ is, by definition of a function, the domain of ⁠${\displaystyle f}$⁠.