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Composition of Functions

The function whose value at x is f ( g ( x ) ) is called the composite of the functions f and g .  The operation that combines f and g to produce the composite is called composition .

Notation:  ( f g ) ( x ) or f ( g ( x ) )

The domain of f ( g ( x ) ) is the set of all x in the domain of g such that g ( x ) is in the domain of f .

Example:

If f ( x ) = x and g ( x ) = 3 x 5 , then find f ( g ( 4 ) ) .

Substituting 4 for x in the definition of the function g , g ( 4 ) = 3 ( 4 ) 5 = 7 . Then, f ( g ( 4 ) ) = f ( 7 )

f ( 7 ) = 7