Editing Function (mathematics) (section)

=== Image and preimage ===
{{Main|Image (mathematics)}}
Let <math>f: X\to Y.</math> The ''image'' under {{mvar|f}} of an element {{mvar|x}} of the domain {{mvar|X}} is {{math|''f''(''x'')}}.<ref name="EOM Function"/> If {{math|''A''}} is any subset of {{math|''X''}}, then the ''image'' of {{mvar|A}} under {{mvar|f}}, denoted {{math|''f''(''A'')}}, is the subset of the codomain {{math|''Y''}} consisting of all images of elements of {{mvar|A}},<ref name="EOM Function"/> that is,

<math display="block">f(A)=\{f(x)\mid x\in A\}.</math>

The ''image'' of {{math|''f''}} is the image of the whole domain, that is, {{math|''f''(''X'')}}.{{r|PCM p.11}} It is also called the [[range of a function|range]] of {{mvar|f}},{{r|EOM Function|T&K Calc p.3|Trench RA pp.30-32|TBB RA pp.A4-A5}} although the term ''range'' may also refer to the codomain.{{r|TBB RA pp.A4-A5|PCM p.11}}<ref name = "standard">''Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology'', p. 15.  ISO 80000-2 (ISO/IEC 2009-12-01)</ref>

On the other hand, the ''[[inverse image]]'' or ''[[preimage]]'' under {{mvar|f}} of an element {{mvar|y}} of the codomain {{mvar|Y}} is the set of all elements of the domain {{math|''X''}} whose images under {{mvar|f}} equal {{mvar|y}}.<ref name="EOM Function"/> In symbols, the preimage of {{mvar|y}} is denoted by <math>f^{-1}(y)</math> and is given by the equation

<math display="block">f^{-1}(y) = \{x \in X \mid f(x) = y\}.</math>

Likewise, the preimage of a subset {{math|''B''}} of the codomain {{math|''Y''}} is the set of the preimages of the elements of {{math|''B''}}, that is, it is the subset of the domain {{math|''X''}} consisting of all elements of {{math|''X''}} whose images belong to {{math|''B''}}.<ref name="EOM Function"/> It is denoted by <math>f^{-1}(B)</math> and is given by the equation

<math display="block">f^{-1}(B) = \{x \in X \mid f(x) \in B\}.</math>

For example, the preimage of <math>\{4, 9\}</math> under the [[square function]] is the set <math>\{-3,-2,2,3\}</math>.

By definition of a function, the image of an element {{math|''x''}} of the domain is always a single element of the codomain. However, the preimage <math>f^{-1}(y)</math> of an element {{mvar|y}} of the codomain may be [[empty set|empty]] or contain any number of elements. For example, if {{mvar|f}} is the function from the integers to themselves that maps every integer to 0, then <math>f^{-1}(0) = \mathbb{Z}</math>.

If <math>f : X\to Y</math> is a function, {{math|''A''}} and {{math|''B''}} are subsets of {{math|''X''}}, and {{math|''C''}} and {{math|''D''}} are subsets of {{math|''Y''}}, then one has the following properties:
* <math>A\subseteq B \Longrightarrow f(A)\subseteq f(B)</math>
* <math>C\subseteq D \Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)</math>
* <math>A \subseteq f^{-1}(f(A))</math>
* <math>C \supseteq f(f^{-1}(C))</math>
* <math>f(f^{-1}(f(A)))=f(A)</math>
* <math>f^{-1}(f(f^{-1}(C)))=f^{-1}(C)</math>

The preimage by {{mvar|f}} of an element {{mvar|y}} of the codomain is sometimes called, in some contexts, the [[fiber (mathematics)|fiber]] of {{math|''y''}} under {{mvar|''f''}}.

If a function {{mvar|f}} has an inverse (see below), this inverse is denoted <math>f^{-1}.</math> In this case <math>f^{-1}(C)</math> may denote either the image by <math>f^{-1}</math> or the preimage by {{mvar|f}} of {{mvar|C}}. This is not a problem, as these sets are equal. The notation <math>f(A)</math> and <math>f^{-1}(C)</math> may be ambiguous in the case of sets that contain some subsets as elements, such as <math>\{x, \{x\}\}.</math> In this case, some care may be needed, for example, by using square brackets <math>f[A], f^{-1}[C]</math> for images and preimages of subsets and ordinary parentheses for images and preimages of elements.