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. 

  1. f [ g(4)] 
  2. 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:

  1. f [ g( x)] 
  2. g[ f ( x)] 
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