 Image and preimage of a function

The image of a function asks which outputs a chosen set of inputs reaches. A preimage asks which inputs reach a chosen set of outputs. Both constructions apply to every function, including one that loses information by sending several inputs to the same output. Their difference is useful whenever a transformation must be followed forward or a condition on its result must be traced back.

Consider the function $f:A\to B$ withEvery input has exactly one output, but zero has two inputs and three has none. The full image of a function is $f(A)=\{0,1,2\}$, a proper subset of the codomain $B$. Selecting inputs $\{a,c\}$ gives image $\{0,1\}$. Selecting outputs $\{0,2\}$ instead gives preimage $\{a,b,d\}$.

The function is not injective, because $f(a)=f(b)$, and is not surjective onto $B$, because three is absent from its image. Restricting only the codomain to $\{0,1,2\}$ makes the same rule surjective; it remains noninjective. This separates the output set actually reached from the target set specified in the definition. The definitions of images of a function and preimages, including the distinction from an inverse function, are given in Richard Hammack, Book of Proof, §12.6: Image and Preimage.