Compositions of Functions
When the input in a function is another function, the result is called a composite function. If
f ( x) = 3 x + 2 and g( x) = 4 x – 5
then f [ g( x)] is a composite function. The statement f [ g( x)] is read “ f of g of x” or “the composition of f with g.” f [ g( x)] can also be written as
The symbol between f and g is a small open circle. When replacing one function with another, be very careful to get the order correct because compositions of functions are not necessarily commutative (as you'll see).
Example 1
If f ( x) = 3 x + 2 and g( x) = 4 x – 5, find each of the following.

f [ g(4)]


f [ g( x)]





From the previous example, you should be convinced that f [ g( x)] and g[ f ( x)] are not (necessarily) equal.
Example 2
If f( x) = 3 x ^{2} + 2 x + 1 and g( x) = 4 x – 5, find each of the following:

f [ g( x)]

g[ f ( x)]

