Editing Function (mathematics) (section)

== General properties ==

This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.

=== Standard functions ===
There are a number of standard functions that occur frequently:
* For every set {{mvar|X}}, there is a unique function, called the '''{{vanchor|empty function}}''', or '''empty map''', from the [[empty set]] to {{mvar|X}}. The graph of an empty function is the empty set.<ref group=note>By definition, the graph of the empty function to {{mvar|X}} is a subset of the Cartesian product {{math|∅ × ''X''}}, and this product is empty.</ref> The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an [[Tuple|ordered triplet]] (or equivalent ones), there is exactly one empty function for each set, thus the empty function <math>\varnothing \to X</math> is not equal to <math>\varnothing \to Y</math> if and only if <math>X\ne Y</math>, although their graphs are both the [[empty set]].
* For every set {{mvar|X}} and every [[singleton set]] {{math|{{mset|''s''}}}}, there is a unique function from {{mvar|X}} to {{math|{{mset|''s''}}}}, which maps every element of {{mvar|X}} to {{mvar|s}}. This is a surjection (see below) unless {{mvar|X}} is the empty set.
* Given a function <math>f: X\to Y,</math> the ''canonical surjection'' of {{mvar|f}} onto its image <math>f(X)=\{f(x)\mid x\in X\}</math> is the function from {{mvar|X}} to {{math|''f''(''X'')}} that maps {{mvar|x}} to {{math|''f''(''x'')}}.
* For every [[subset]] {{mvar|A}} of a set {{mvar|X}}, the [[inclusion map]] of {{mvar|A}} into {{mvar|X}} is the injective (see below) function that maps every element of {{mvar|A}} to itself.
* The [[identity function]] on a set {{mvar|X}}, often denoted by {{math|id<sub>''X''</sub>}}, is the inclusion of {{mvar|X}} into itself.

=== Function composition ===
{{Main|Function composition}}

Given two functions <math>f: X\to Y</math> and <math>g: Y\to Z</math> such that the domain of {{mvar|g}} is the codomain of {{mvar|f}}, their ''composition'' is the function <math>g \circ f: X \rightarrow Z</math> defined by

<math display="block">(g \circ f)(x) = g(f(x)).</math>

That is, the value of <math>g \circ f</math> is obtained by first applying {{math|''f''}} to {{math|''x''}} to obtain {{math|1=''y'' = ''f''(''x'')}} and then applying {{math|''g''}} to the result {{mvar|y}} to obtain {{math|1=''g''(''y'') = ''g''(''f''(''x''))}}. In this notation, the function that is applied first is always written on the right.

The composition <math>g\circ f</math> is an [[operation (mathematics)|operation]] on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both <math>g \circ f</math> and <math>f \circ g</math> satisfy these conditions, the composition is not necessarily [[commutative property|commutative]], that is, the functions <math>g \circ f</math> and <math> f \circ g</math> need not be equal, but may deliver different values for the same argument. For example, let {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} and {{math|1=''g''(''x'') = ''x'' + 1}}, then <math>g(f(x))=x^2+1</math> and <math> f(g(x)) = (x+1)^2</math> agree just for <math>x=0.</math>

The function composition is [[associative property|associative]] in the sense that, if one of <math>(h\circ g)\circ f</math> and <math>h\circ (g\circ f)</math> is defined, then the other is also defined, and they are equal, that is, <math>(h\circ g)\circ f = h\circ (g\circ f).</math> Therefore, it is usual to just write <math>h\circ g\circ f.</math>

The [[identity function]]s <math>\operatorname{id}_X</math> and <math>\operatorname{id}_Y</math> are respectively a [[right identity]] and a [[left identity]] for functions from {{mvar|X}} to {{mvar|Y}}. That is, if {{mvar|f}} is a function with domain {{mvar|X}}, and codomain {{mvar|Y}}, one has
<math>f\circ \operatorname{id}_X = \operatorname{id}_Y \circ f = f.</math>

<gallery widths="250" heights="300">
File:Function machine5.svg|A composite function ''g''(''f''(''x'')) can be visualized as the combination of two "machines".
File:Example for a composition of two functions.svg|A simple example of a function composition
File:Compfun.svg|Another composition. In this example, {{math|1=(''g'' ∘ ''f'' )(c) = #}}.
</gallery>

=== 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.

=== Injective, surjective and bijective functions ===
{{main|Bijection, injection and surjection}}

Let <math>f : X\to Y</math> be a function.

The function {{mvar|f}} is ''[[injective function|injective]]'' (or ''one-to-one'', or is an ''injection'') if {{math|''f''(''a'') ≠ ''f''(''b'')}} for every two different elements {{math|''a''}} and {{mvar|''b''}} of {{mvar|X}}.<ref name="PCM p.11">{{Princeton Companion to Mathematics|p=11}}</ref><ref name="EOM Injection">{{eom |title=Injection |oldid=30986 |first=O. A. |last=Ivanova |mode=cs1}}</ref> Equivalently, {{mvar|f}} is injective if and only if, for every <math>y\in Y,</math> the preimage <math>f^{-1}(y)</math> contains at most one element. An empty function is always injective. If {{mvar|X}} is not the empty set, then {{mvar|f}} is injective if and only if there exists a function <math>g: Y\to X</math> such that <math>g\circ f=\operatorname{id}_X,</math> that is, if {{mvar|f}} has a [[left inverse function|left inverse]].<ref name="EOM Injection"/> ''Proof'': If {{mvar|f}} is injective, for defining {{mvar|g}}, one chooses an element <math>x_0</math> in {{mvar|X}} (which exists as {{mvar|X}} is supposed to be nonempty),<ref group=note>The [[axiom of choice]] is not needed here, as the choice is done in a single set.</ref> and one defines {{mvar|g}} by <math>g(y)=x</math> if <math>y=f(x)</math> and <math>g(y)=x_0</math> if <math>y\not\in f(X).</math> Conversely, if <math>g\circ f=\operatorname{id}_X,</math> and <math>y=f(x),</math>  then <math>x=g(y),</math> and thus <math>f^{-1}(y)=\{x\}.</math>

The function {{mvar|f}} is ''[[surjective]]'' (or ''onto'', or is a ''surjection'') if its range <math>f(X)</math> equals its codomain <math>Y</math>, that is, if, for each element <math>y</math> of the codomain, there exists some element <math>x</math> of the domain such that <math>f(x) = y</math> (in other words, the preimage <math>f^{-1}(y)</math> of every <math>y\in Y</math> is nonempty).<ref name="PCM p.11"/><ref name="EOM Surjection">{{eom |title=Surjection |oldid=35689 |author-first=O.A. |author-last=Ivanova|mode=cs1}}</ref> If, as usual in modern mathematics, the [[axiom of choice]] is assumed, then {{mvar|f}} is surjective if and only if there exists a function <math>g: Y\to X</math> such that <math>f\circ g=\operatorname{id}_Y,</math> that is, if {{mvar|f}} has a [[right inverse function|right inverse]].<ref name="EOM Surjection"/> The axiom of choice is needed, because, if {{mvar|f}} is surjective, one defines {{mvar|g}} by <math>g(y)=x,</math> where <math>x</math> is an ''arbitrarily chosen'' element of <math>f^{-1}(y).</math>

The function {{mvar|f}} is ''[[bijective]]'' (or is a ''bijection'' or a ''one-to-one correspondence'') if it is both injective and surjective.<ref name="PCM p.11"/><ref name="EOM Bijection">{{eom |title=Bijection |oldid=30987 |author-first=O.A. |author-last=Ivanova|mode=cs1}}</ref> That is, {{mvar|f}} is bijective if, for every <math>y\in Y,</math> the preimage <math>f^{-1}(y)</math> contains exactly one element. The function {{mvar|f}} is bijective if and only if it admits an [[inverse function]], that is, a function <math>g : Y\to X</math> such that <math>g\circ f=\operatorname{id}_X</math> and <math>f\circ g=\operatorname{id}_Y.</math><ref name="EOM Bijection"/> (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).

Every function <math>f: X\to Y</math> may be [[factorization|factorized]] as the composition <math>i\circ s</math> of a surjection followed by an injection, where {{mvar|s}} is the canonical surjection of {{mvar|X}} onto {{math|''f''(''X'')}} and {{mvar|i}} is the canonical injection of {{math|''f''(''X'')}} into {{mvar|Y}}. This is the ''canonical factorization'' of {{mvar|f}}.

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the [[Nicolas Bourbaki|Bourbaki group]] and imported into English.<ref>{{Cite web |last=Hartnett |first=Kevin |date=9 November 2020 |title=Inside the Secret Math Society Known Simply as Nicolas Bourbaki |url=https://www.quantamagazine.org/inside-the-secret-math-society-known-as-nicolas-bourbaki-20201109/ |access-date=2024-06-05 |website=Quanta Magazine}}</ref> As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function.  Also, the statement "{{math|''f''}} maps {{math|''X''}} ''onto'' {{math|''Y''}}" differs from "{{math|''f''}}  maps {{math|''X''}} ''into'' {{math|''B''}}", in that the former implies that {{math|''f''}} is surjective, while the latter makes no assertion about the nature of {{math|''f''}}. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.

=== Restriction and extension <span class="anchor" id="Restrictions and extensions"></span> ===
<!-- This section is linked from [[Subgroup]], [[Restriction]], [[Quadratic form]] -->
{{main|Restriction (mathematics)}}
If <math>f : X \to Y</math> is a function and {{math|''S''}} is a subset of {{math|''X''}}, then the ''restriction'' of <math>f</math> to ''S'', denoted <math>f|_S</math>, is the function from {{math|''S''}} to {{math|''Y''}} defined by

<math display="block">f|_S(x) = f(x)</math>

for all {{math|''x''}} in {{math|''S''}}. Restrictions can be used to define partial [[inverse function]]s: if there is a [[subset]] {{math|''S''}} of the domain of a function <math>f</math> such that <math>f|_S</math> is injective, then the canonical surjection of <math>f|_S</math> onto its image <math>f|_S(S) = f(S)</math> is a bijection, and thus has an inverse function from <math>f(S)</math> to {{math|''S''}}. One application is the definition of [[inverse trigonometric functions]]. For example, the [[cosine]] function is injective when restricted to the [[interval (mathematics)|interval]] {{closed-closed|0, ''π''}}. The image of this restriction is the interval {{closed-closed|−1, 1}}, and thus the restriction has an inverse function from {{closed-closed|−1, 1}} to {{closed-closed|0, ''π''}}, which is called [[arccosine]] and is denoted {{math|arccos}}.

Function restriction may also be used for "gluing" functions together. Let <math display="inline"> X=\bigcup_{i\in I}U_i</math> be the decomposition of {{mvar|X}} as a [[set union|union]] of subsets, and suppose that a function <math>f_i : U_i \to Y</math> is defined on each <math>U_i</math> such that for each pair <math>i, j</math> of indices, the restrictions of <math>f_i</math> and <math>f_j</math> to <math>U_i \cap U_j</math> are equal. Then this defines a unique function <math>f : X \to Y</math> such that <math>f|_{U_i} = f_i</math> for all {{mvar|i}}. This is the way that functions on [[manifold]]s are defined.

An ''extension'' of a function {{mvar|f}} is a function {{mvar|g}} such that {{mvar|f}} is a restriction of {{mvar|g}}. A typical use of this concept is the process of [[analytic continuation]], that allows extending functions whose domain is a small part of the [[complex plane]] to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying [[homography|homographies]] of the [[real line]]. A ''homography'' is a function <math>h(x)=\frac{ax+b}{cx+d}</math> such that {{math|''ad'' − ''bc'' ≠ 0}}. Its domain is the set of all [[real number]]s different from <math>-d/c,</math> and its image is the set of all real numbers different from <math>a/c.</math> If one extends the real line to the [[projectively extended real line]] by including {{math|∞}}, one may extend {{mvar|h}} to a bijection from the extended real line to itself by setting <math>h(\infty)=a/c</math> and <math>h(-d/c)=\infty</math>.