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 yvalue
answers) of one function becomes
the domain (the xvalues) 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 xvalue. 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
TI83+/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.)
