
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. 
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 

Click on the check boxes in manipulative 1 to show the compositions of functions. 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  X 
Y  Z 
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