Editing Function (mathematics) (section)

=== 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>.