# Composite Function

Pronunciation: /kəmˈpɒz.ɪt ˈfʌŋk.ʃən/ Explain

 Figure 1: Composite function

A composite function is a function that is made up of two other functions that takes as input of one function the output of the other function[3][4]. To write the composition of functions f(x) and g(x), write f(x) ° g(x) or f(g(x)).

A composite function is sometimes called a compound function. The act of composing functions is called composition of functions.

### Properties of composite functions

Some properties of composite functions are:

PropertyEquationDescription
Associative f ° (g ° h(x)) = (f ° g) ° h(x) Composition of functions is associative.
Commutative f ° g(x)g ° f(x) Composition of functions is not commutative.
Inverse functions f ° f-1(x) = x The composition of a function and its inverse is the identity function (f(x) = x).
Linear functions f(x) = ax + b, g(x) = cx + d, f(g(x)) = acx + ad+b The composition of two linear functions is also linear.
PolynomialsThe degree of the composition of two polynomials is the product of the degrees of the two functions.
Table 1: Properties of composite functions

 Check one check box on the left and one on the right to see the two functions and their composite. Does f(g(x)) always equal g(f(x))? Manipulative 1 - Composite Functions Created with GeoGebra.

### References

1. McAdams, David E.. All Math Words Dictionary, composite function. 2nd Classroom edition 20150108-4799968. pg 40. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. composite function. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 6/25/2018. http://www.merriam-webster.com/dictionary/composite function. Buy the book
3. Goldrei, D.C.. Classic Set Theory: For Guided Independent Study. pg 5. Chapman & Hall Mathematics. July 1 1996. Last Accessed 6/25/2018. Buy the book
4. R. K. Singh, J. S. Manhas. Composition Operators on Function Spaces. North-Holland. November 1 1993. Last Accessed 6/25/2018. Buy the book
5. Carl C. Cowen, Barbara I. Maccluer. Composition Operators on Space of Analytic Functions. CRC-Press. June 30 1995. Last Accessed 6/25/2018. Buy the book

• McAdams, David E.. Composite. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/c/composite.html.

McAdams, David E. Composite Function. 4/14/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/compositefunction.html.

### Revision History

4/14/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
5/5/2011: Initial version. (McAdams, David E.)

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