Editing Image (mathematics) (section)

{{Short description|Set of the values of a function}}
[[File:FunctionMappingPersonToFavoriteFood.png|350px|thumb|right|For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.]]
{{other uses|Image (disambiguation)}}

In [[mathematics]], for a function <math>f: X \to Y</math>, the '''image''' of an input value <math>x</math> is the single output value produced by <math>f</math> when passed  <math>x</math>. The '''preimage''' of an output value <math>y</math> is the set of input values that produce <math>y</math>.

More generally, evaluating <math>f</math> at each [[Element (mathematics)|element]] of a given subset <math>A</math> of its [[Domain of a function|domain]] <math>X</math> produces a set, called the "'''image''' of <math>A</math> under (or through) <math>f</math>". Similarly, the '''inverse image''' (or '''preimage''') of a given subset <math>B</math> of the [[codomain]] <math>Y</math> is the set of all elements of <math>X</math> that map to a member of <math>B.</math>

The '''image''' of the function <math>f</math> is the set of all output values it may produce, that is, the image of <math>X</math>. The '''preimage''' of <math>f</math> is the preimage of the codomain <math>Y</math>. Because it always equals <math>X</math> (the domain of <math>f</math>), it is rarely used.

Image and inverse image may also be defined for general [[Binary relation#Operations|binary relations]], not just functions.