Composite Function

Pronunciation: /kəmˈpɒz.ɪt ˈfʌŋk.ʃən/ Explain

On top are two boxes and three arrows. The box on the left is labeled g(x). The box on the right is labeled f(x). An arrow labeled input points to the box labeled g(x). An arrow points from the box labeled g(x) to the box labeled f(x). An arrow labeled output points out from the box labeled f(x). On the bottom there is one long box and two arrows. The box is labeled f(x)?g(x) and f(g(x)). The arrow going into the box is labeled input and the arrow leading from the box is labeled output.
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:

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. Encyclopedia Britannica. Merriam-Webster. Last Accessed 6/25/2018. 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

More Information

  • McAdams, David E.. Composite. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018.

Cite this article as:

McAdams, David E. Composite Function. 4/14/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

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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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