Mathematics → Subject Test → 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 ) = 10^e–7 Then ( f ∘ g ) ( x )=?
Solution
( f ∘ g ) ( x ) = f ( g ( x ) )
= f ( 10^e–7 )
= ln ( log 10^e–7 )
= ln e^–7
= –7 Answer

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