Composite Function
Pronunciation: /kəmˈpɒz ɪt ˈfʌŋk ʃən/ Explain
 Figure 1: Composite function 

A composite function is a
function
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:
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
 composite function. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 8/6/2018. http://www.merriamwebster.com/dictionary/composite function.
 Goldrei, D.C.. Classic Set Theory: For Guided Independent Study. pg 5. Chapman & Hall Mathematics. July 1 1996. Last Accessed 8/6/2018. Buy the book
 R. K. Singh, J. S. Manhas. Composition Operators on Function Spaces. NorthHolland. November 1 1993. Buy the book
 Carl C. Cowen, Barbara I. Maccluer. Composition Operators on Space of Analytic Functions. CRCPress. June 30 1995. 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. 6/29/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/compositefunction.html.
Image Credits
Revision History
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (
McAdams, David E.)