Function composition - Revision history

~2025-31850-11: I don’t think this is a thing, but in any case it should not be in the lead if it’s not in the body

2025-12-29T12:58:35Z

I don’t think this is a thing, but in any case it should not be in the lead if it’s not in the body

← Previous revision		Revision as of 12:58, 29 December 2025
Line 1:		Line 1:
	{{short description\|Operation on mathematical functions}}		{{short description\|Operation on mathematical functions}}
	{{about\|the mathematical ~~concept\|the computer science~~ concept\|Function composition (computer science)}}		{{about-distinguish\|the mathematical concept\|Function composition (computer science)}}
	{{redirect-distinguish\|Ring operator\|operator ring\|operator assistance}}		{{redirect-distinguish\|Ring operator\|operator ring\|operator assistance}}
	{{redirects\|∘\|the character\|Degree symbol#Lookalikes}}		{{redirects\|∘\|the character\|Degree symbol#Lookalikes}}
Line 6:		Line 6:
	{{Functions}}		{{Functions}}

	In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x~~) :=~~ (g \circ f) (x) = g(f(x))</math>. ~~Thus~~, the function {{math\|''g''}} is [[function application\|applied]] after applying {{math\|''f''}} to {{math\|''x''}}. <math>(g \circ f)</math> is pronounced "~~the composition of~~ {{math\|''g''}} ~~and~~ {{math\|''f''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>		In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>f \circ g</math>. When the composite function <math>f \circ g</math> is evaluated at an input <math>x</math>, the result is <math>(f \circ g)(x) = f(g(x))</math>. That is, the function {{math\|''f''}} is [[function application\|applied]] after applying {{math\|''g''}} to {{math\|''x''}}. The symbol <math>(f \circ g)</math> is pronounced "{{math\|''f''}} of {{math\|''g''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>

	'''Reverse composition''' applies the operation in the opposite order, applying <math>f</math> first and <math>g</math> second. Intuitively, reverse composition is a chaining process in which the output of function {{math\|''f''}} feeds the input of function {{math\|''g''}}.

	The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].		The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].

91.215.141.68: \mapsto has a different purpose: (Function (mathematics)) "One may write x ↦ y instead of y=f(x), where the symbol ↦ (read 'maps to') is used to specify where a particular element x in the domain is mapped to by f. This allows the definition of a function without naming. For example, the square function is the function x ↦ x^2." https://en.wikipedia.org/w/index.php?title=Function_composition&diff=prev&oldid=1241337555

2025-08-17T13:43:10Z

\mapsto has a different purpose: (Function (mathematics)) "One may write x ↦ y instead of y=f(x), where the symbol ↦ (read 'maps to') is used to specify where a particular element x in the domain is mapped to by f. This allows the definition of a function without naming. For example, the square function is the function x ↦ x^2." https://en.wikipedia.org/w/index.php?title=Function_composition&diff=prev&oldid=1241337555

← Previous revision		Revision as of 13:43, 17 August 2025
Line 8:		Line 8:
	In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x) := (g \circ f) (x) = g(f(x))</math>. Thus, the function {{math\|''g''}} is [[function application\|applied]] after applying {{math\|''f''}} to {{math\|''x''}}. <math>(g \circ f)</math> is pronounced "the composition of {{math\|''g''}} and {{math\|''f''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>		In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x) := (g \circ f) (x) = g(f(x))</math>. Thus, the function {{math\|''g''}} is [[function application\|applied]] after applying {{math\|''f''}} to {{math\|''x''}}. <math>(g \circ f)</math> is pronounced "the composition of {{math\|''g''}} and {{math\|''f''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>

	'''Reverse composition'''~~, sometimes denoted <math>f \mapsto g</math> ,~~ applies the operation in the opposite order, applying <math>f</math> first and <math>g</math> second. Intuitively, reverse composition is a chaining process in which the output of function {{math\|''f''}} feeds the input of function {{math\|''g''}}.		'''Reverse composition''' applies the operation in the opposite order, applying <math>f</math> first and <math>g</math> second. Intuitively, reverse composition is a chaining process in which the output of function {{math\|''f''}} feeds the input of function {{math\|''g''}}.

	The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].		The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].

imported>Jochen Burghardt: /* Composition monoids */ suggest to link the article about the jargon term "up to", where "up to isomorphism" is one of the examples (see up to#Group theory, linking there directly could be an WP:EASTEREGG); rm adjacent link isomorphism per MOS:SPECIFICLINK;

2025-02-25T08:50:31Z

Composition monoids: suggest to link the article about the jargon term "up to", where "up to isomorphism" is one of the examples (see up to#Group theory, linking there directly could be an WP:EASTEREGG); rm adjacent link isomorphism per MOS:SPECIFICLINK;

Show changes

← Previous revision		Revision as of 12:58, 29 December 2025
Line 1:		Line 1:
	{{short description\|Operation on mathematical functions}}		{{short description\|Operation on mathematical functions}}
	{{about\|the mathematical ~~concept\|the computer science~~ concept\|Function composition (computer science)}}		{{about-distinguish\|the mathematical concept\|Function composition (computer science)}}
	{{redirect-distinguish\|Ring operator\|operator ring\|operator assistance}}		{{redirect-distinguish\|Ring operator\|operator ring\|operator assistance}}
	{{redirects\|∘\|the character\|Degree symbol#Lookalikes}}		{{redirects\|∘\|the character\|Degree symbol#Lookalikes}}
Line 6:		Line 6:
	{{Functions}}		{{Functions}}

	In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x~~) :=~~ (g \circ f) (x) = g(f(x))</math>. ~~Thus~~, the function {{math\|''g''}} is [[function application\|applied]] after applying {{math\|''f''}} to {{math\|''x''}}. <math>(g \circ f)</math> is pronounced "~~the composition of~~ {{math\|''g''}} ~~and~~ {{math\|''f''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>		In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>f \circ g</math>. When the composite function <math>f \circ g</math> is evaluated at an input <math>x</math>, the result is <math>(f \circ g)(x) = f(g(x))</math>. That is, the function {{math\|''f''}} is [[function application\|applied]] after applying {{math\|''g''}} to {{math\|''x''}}. The symbol <math>(f \circ g)</math> is pronounced "{{math\|''f''}} of {{math\|''g''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>

	'''Reverse composition''' applies the operation in the opposite order, applying <math>f</math> first and <math>g</math> second. Intuitively, reverse composition is a chaining process in which the output of function {{math\|''f''}} feeds the input of function {{math\|''g''}}.

	The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].		The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].

← Previous revision		Revision as of 13:43, 17 August 2025
Line 8:		Line 8:
	In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x) := (g \circ f) (x) = g(f(x))</math>. Thus, the function {{math\|''g''}} is [[function application\|applied]] after applying {{math\|''f''}} to {{math\|''x''}}. <math>(g \circ f)</math> is pronounced "the composition of {{math\|''g''}} and {{math\|''f''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>		In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)\|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x) := (g \circ f) (x) = g(f(x))</math>. Thus, the function {{math\|''g''}} is [[function application\|applied]] after applying {{math\|''f''}} to {{math\|''x''}}. <math>(g \circ f)</math> is pronounced "the composition of {{math\|''g''}} and {{math\|''f''}}".<ref>{{Cite web \|title=Composition of Functions \|url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php \|access-date=2025-02-07 \|website=nool.ontariotechu.ca \|language=en}}</ref>

	'''Reverse composition'''~~, sometimes denoted <math>f \mapsto g</math> ,~~ applies the operation in the opposite order, applying <math>f</math> first and <math>g</math> second. Intuitively, reverse composition is a chaining process in which the output of function {{math\|''f''}} feeds the input of function {{math\|''g''}}.		'''Reverse composition''' applies the operation in the opposite order, applying <math>f</math> first and <math>g</math> second. Intuitively, reverse composition is a chaining process in which the output of function {{math\|''f''}} feeds the input of function {{math\|''g''}}.

	The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].		The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties\|associativity]].