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:
Property  Equation  Description 
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. 
Polynomials   The 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
 McAdams, David E.. All Math Words Dictionary, composite function. 2nd Classroom edition 201501084799968. pg 40. Life is a Story Problem LLC. January 8, 2015. Buy the book
 composite function. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 6/25/2018. http://www.merriamwebster.com/dictionary/composite function. Buy the book
 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
 R. K. Singh, J. S. Manhas. Composition Operators on Function Spaces. NorthHolland. November 1 1993. Last Accessed 6/25/2018. Buy the book
 Carl C. Cowen, Barbara I. Maccluer. Composition Operators on Space of Analytic Functions. CRCPress. June 30 1995. Last Accessed 6/25/2018. Buy the book
More Information
 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.
Cite this article as:
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.
Image Credits
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.)