Composite Function

Pronunciation: /kəmˈpɒz ɪt ˈfʌŋk ʃən/ ?

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 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:

Associativef°(g°h(x)) = (f°g)°h(x)Composition of functions is associative.
Commutativef°g(x) ≠ g°f(x)Composition of functions is not commutative.
Inverse functionsf°f-1(x)= xThe composition of a function and its inverse is the identity function (f(x)=x).
Linear functionsf(x) = ax + b, g(x) = cx + d, f(g(x)) = acx + ad+bThe 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

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Manipulative 1: Composite functions

Click on the check boxes in manipulative 1 to show the compositions of functions.


  1. composite function. Encyclopedia Britannica. (Accessed: 2010-01-26). function.
  2. Goldrei, D.C.. Classic Set Theory: For Guided Independent Study, pg 5. Chapman & Hall Mathematics, July 1, 1996.
  3. R. K. Singh, J. S. Manhas. Composition Operators on Function Spaces. North-Holland, November 1, 1993.
  4. Carl C. Cowen, Barbara I. Maccluer. Composition Operators on Space of Analytic Functions. CRC-Press, June 30, 1995.

More Information

  • McAdams, David. Composite. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-01-26.
  • Staple, Elizabeth. Composition of Functions. Purple Math. 2010-01-25.

Printed Resources

Cite this article as:

Composite Function. 2010-01-28. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

2010-01-26: Initial version (McAdams, David.)

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