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Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in April 2021. This work is licensed under a Creative Commons BY 4.0 International License.

Composition of Functions

In mathematics, a Function Composition represented by a circle is an operation that combine two functions f(x) and g(x) to produce a function h(x):
In a composition of functions, the domain of the first function f(x) becomes g(x). Therefore, in order to find the final function h(x), one has to replace x in f(x) by g(x).


Example: For  and , find 
In order to solve this problem, we have to replace x in f(x) by g(x):

Example: Find the composite function for and 

Example: Find the domain of the composite functions 

a) First, we need to find the composite function by replacing x by f(x):
the domain of the function is all real numbers.
b) We find the composite function:
the domain is 
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