Composition of Functions
(f o g)(x)
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The term "composition of functions" (or "composite function") refers
 to the combining of functions in a manner where the output from one function
becomes the input for the next function. 

In math terms, the range (the y-value answers) of one function becomes
the domain (the x-values) of the next function.

The notation used for composition is:

and is read "f composed with g of x" or "f of g of x".

Notice how the letters stay in the same order in each expression for the composition.
 f
(g(x)) clearly tells you to start with function g (innermost parentheses are done first).


Composition of functions can be thought of as a series of taxicab rides for your values. 
  The example below shows functions f and g working together to create the
composition

Note: 
The starting domain for function g is being limited to the four values 1, 2, 3 and 4 for this example.

In the example above, you can see what is happening to the individual elements
throughout the composition.  Now, suppose that we wish to write our composition as an algebraic expression.

1.  Substitute the expression for function g (in this case 2x) for g(x) in the composition.  This will clearly show you the order of the substitutions that will need to be made.

2.  Now, substitute this expression (2x) into function f in place of the x-value.  Perform any needed simplifications (none needed in this example).

 

You will find that the concept of composite functions is widely used.
  For example, you are often using composite functions when you are
evaluating expressions on a calculator.

Evaluate:
                        

 

When entering this computation on the calculator, you will press the button to square a value and the button to raise a value to any power (exponentiation).  The buttons represent the functions for squaring and exponentiation.   This problem is dealing with the composition of these two functions.  The problem could be represented as  evaluated at x = 3.6 where and  .
(The order of entry will vary depending upon the model of calculator.)

How to use your
TI-83+/84+ graphing calculator
 with composition of functions.
Click here.


Examples:

1.  Given the functions and , find  a.) and  b.)

     Answer:  a.) 

                   b.)  

Notice that and do not necessarily yield the same answer.
Composition of functions is not commutative.

 

2.  Given the functions and , find  a.) and  b.)

     Answer:  a.)   = h(p(3)) where p(3) gives an answer of 5
                          and h(5) then gives an answer of 25.
                          The answer is 25.

                   b.)