Composition of Functions

Definition

Given functions f and g, the composition of f with g is denoted by
f g, is the function defined by

( f g ) ( x ) = f ( g ( x ) )

Similarly, the composition of g with f is denoted by g f, is the function defined by

( g f ) ( x ) = g ( f ( x ) )

So, compositions are evaluated by plugging the second function into the first function.

NOTE:

( f g ) ( x ) ( g f ) ( x )

Practice Problems

Example 1. Given f ( x ) = x2 + 1, and g ( x ) = 1 - x, find the following:
(a) ( f g ) ( x )
(b) ( g f ) ( x )
Solution
(a) ( f g ) ( x )
( f g ) ( x ) = f ( g ( x ) )
= f ( 1 - x )
= ( 1 - x )2 + 1
= 1 - 2x + x2 + 1
= x2 - 2 x + 2 Answer

(b) ( g f ) ( x )
Solution
( g f ) ( x )
( g f ) ( x ) = g ( f ( x ) )
= g ( x2 + 1 )
= 1 - x2 + 1
= 2 - x2 Answer

Example 2. If f ( x ) = ln ( log x ) and g ( x ) = 10e-7 Then ( f g ) ( x )=?
Solution
( f g ) ( x ) = f ( g ( x ) )
= f ( 10e-7 )
= ln ( log 10e-7 )
= ln e-7
= -7 Answer

NOTE: Here we use the following facts:
log 10 = 1 and
ln e = 1