Composition of Functions - Revision history

Khanh at 21:55, 20 January 2022

2022-01-20T21:55:29Z

Lila: /* References */

2021-10-21T20:39:07Z

References

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2021-09-20T22:51:26Z

Resources

Lila at 22:49, 20 September 2021

2021-09-20T22:49:50Z

Lila: /* Properties */

2021-09-20T22:46:49Z

Properties

Khanh: Created page with "In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is..."

2021-09-16T02:40:42Z

Created page with "In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is..."

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In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.

Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g.

The composition of functions is a special case of the composition of relations, sometimes also denoted by <math>\circ</math> .[1] As a result, all properties of composition of relations are true of composition of functions, though the composition of functions has some additional properties.

Composition of functions is different from multiplication of functions, and has quite different properties; in particular, composition of functions is not commutative.

==Examples==
[[File:Example for a composition of two functions.svg|thumb|Concrete example for the composition of two functions.]]
* Composition of functions on a finite set: If {{math|1=''f'' = {(1, 1), (2, 3), (3, 1), (4, 2)} }}, and {{math|1=''g'' = {(1, 2), (2, 3), (3, 1), (4, 2)} }}, then {{math|1=''g'' ∘ ''f'' = {(1, 2), (2, 1), (3, 2), (4, 3)} }}, as shown in the figure.
* Composition of functions on an infinite set: If {{math|''f'': ℝ → ℝ}} (where {{math|ℝ}} is the set of all real numbers) is given by {{math|1=''f''(''x'') = 2''x'' + 4}} and {{math|''g'': ℝ → ℝ}} is given by {{math|1=''g''(''x'') = ''x''3}}, then:
:{{math|1=(''f'' ∘ ''g'')(''x'') = ''f''(''g''(''x'')) = ''f''(''x''3) = 2''x''3 + 4}}, and
:{{math|1=(''g'' ∘ ''f'')(''x'') = ''g''(''f''(''x'')) = ''g''(2''x'' + 4) = (2''x'' + 4)3}}.
* If an airplane's altitude at time {{mvar|t}} is {{math|''a''(''t'')}}, and the air pressure at altitude {{mvar|x}} is {{math|''p''(''x'')}}, then {{math|(''p'' ∘ ''a'')(''t'')}} is the pressure around the plane at time {{mvar|t}}.

==Properties==
The composition of functions is always associative—a property inherited from the composition of relations.That is, if {{mvar|f}}, {{mvar|g}}, and {{mvar|h}} are composable, then {{math|1=''f'' ∘ (''g'' ∘ ''h'') = (''f'' ∘ ''g'') ∘ ''h''}}. Since the parentheses do not change the result, they are generally omitted.

In a strict sense, the composition {{math|1=''g'' ∘ ''f''}} is only meaningful if the codomain of {{mvar|f}} equals the domain of {{mvar|g}}; in a wider sense, it is sufficient that the former be a [[subset]] of the latter. Moreover, it is often convenient to tacitly restrict the domain of {{mvar|f}}, such that {{mvar|f}} produces only values in the domain of {{mvar|g}}. For example, the composition {{math|1=''g'' ∘ ''f''}} of the functions {{math|''f'' : real number|ℝ → interval (mathematics)#Infinite endpoints|(−∞,+9] }} defined by {{math|1=''f''(''x'') = 9 − ''x''2}} and {{math|''g'' : interval (mathematics)#Infinite endpoints|[0,+∞) → ℝ}} defined by <math>g(x) = \sqrt x</math> can be defined on the [[interval (mathematics)|interval]] {{math|[−3,+3]}}.

[[Image:Absolute value composition.svg|thumb|upright=1|Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition.]]
The functions {{mvar|g}} and {{mvar|f}} are said to commute with each other if {{math|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, {{math|1={{abs|''x''}} + 3 = {{abs|''x'' + 3}}}} only when {{math|''x'' ≥ 0}}. The picture shows another example.

The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that {{math|1=(''f'' ∘ ''g'')−1 = ''g''−1∘ ''f''−1}}.

==Resources==
[https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-compositions-of-functions/ Compositions of Functions], Lumen Learning

==References==
* "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-28.
* Velleman, Daniel J. (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 232. ISBN 978-1-139-45097-3.
* "3.4: Composition of Functions". Mathematics LibreTexts. 2020-01-16. Retrieved 2020-08-28.
* Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28.
* Rodgers, Nancy (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp. 359–362. ISBN 978-0-471-37122-9.

@@ Line 29: / Line 29: @@
 * [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-compositions-of-functions/ Compositions of Functions], Lumen Learning
 * [https://www.mathsisfun.com/sets/functions-composition.html Composition of Functions], Math Is Fun
 == Licensing ==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Function_composition Function composition, Wikipedia] under a CC BY-SA license

← Older revision		Revision as of 20:39, 21 October 2021
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	* Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28.		* Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28.
	* Rodgers, Nancy (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp. 359–362. ISBN 978-0-471-37122-9.		* Rodgers, Nancy (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp. 359–362. ISBN 978-0-471-37122-9.
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Function_composition Function composition, Wikipedia] under a CC BY-SA license

@@ Line 21: / Line 21: @@
 [[Image:Absolute value composition.svg|thumb|upright=1|Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition.]]
-The functions {{mvar|g}} and {{mvar|f}} are said to commute with each other if {{math|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, {{math|1={{abs|''x''}} + 3 = {{abs|''x'' + 3}}}} only when {{math|''x'' ≥ 0}}. The picture shows another example.
+The functions {{mvar|g}} and {{mvar|f}} are said to commute with each other if {{math|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, <math> |x| + 3 = |x + 3| </math> only when <math> x \ge 0 </math>. The picture shows another example.
 The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that {{math|1=(''f'' ∘ ''g'')<sup>−1</sup> = ''g''<sup>−1</sup>∘ ''f''<sup>−1</sup>}}.

@@ Line 18: / Line 18: @@
 The composition of functions is always associative—a property inherited from the composition of relations.That is, if {{mvar|f}}, {{mvar|g}}, and {{mvar|h}} are composable, then {{math|1=''f'' ∘ (''g'' ∘ ''h'') = (''f'' ∘ ''g'') ∘ ''h''}}. Since the parentheses do not change the result, they are generally omitted.
-In a strict sense, the composition {{math|1=''g'' ∘ ''f''}} is only meaningful if the codomain of {{mvar|f}} equals the domain of {{mvar|g}}; in a wider sense, it is sufficient that the former be a [[subset]] of the latter. Moreover, it is often convenient to tacitly restrict the domain of {{mvar|f}}, such that {{mvar|f}} produces only values in the domain of {{mvar|g}}. For example, the composition {{math|1=''g'' ∘ ''f''}} of the functions {{math|''f'' : real number|ℝ → interval (mathematics)#Infinite endpoints|(−∞,+9] }} defined by {{math|1=''f''(''x'') = 9 − ''x''<sup>2</sup>}} and {{math|''g'' : interval (mathematics)#Infinite endpoints|[0,+∞) → ℝ}} defined by <math>g(x) = \sqrt x</math> can be defined on the [[interval (mathematics)|interval]] {{math|[−3,+3]}}.
+In a strict sense, the composition {{math|1=''g'' ∘ ''f''}} is only meaningful if the codomain of {{mvar|f}} equals the domain of {{mvar|g}}; in a wider sense, it is sufficient that the former be a subset of the latter. Moreover, it is often convenient to tacitly restrict the domain of {{mvar|f}}, such that {{mvar|f}} produces only values in the domain of {{mvar|g}}. For example, the composition {{math|1=''g'' ∘ ''f''}} of the functions {{math|''f'' : real number|ℝ → interval (mathematics)#Infinite endpoints|(−∞,+9] }} defined by {{math|1=''f''(''x'') = 9 − ''x''<sup>2</sup>}} and {{math|''g'' : interval (mathematics)#Infinite endpoints|[0,+∞) → ℝ}} defined by <math>g(x) = \sqrt x</math> can be defined on the interval {{math|[−3,+3]}}.
 [[Image:Absolute value composition.svg|thumb|upright=1|Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition.]]

@@ Line 26: / Line 26: @@
 ==Resources==
-[https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-compositions-of-functions/ Compositions of Functions], Lumen Learning
+* [https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:composing/v/function-composition Intro to Composing Functions], Khan Academy
 ==References==