# 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. 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.

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. https://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. https://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.)