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Operations on Functions

Functions with overlapping domains can be added, subtracted, multiplied and divided.  If   f ( x ) and g ( x ) are two functions, then for all x in the domain of both functions the sum, difference, product and quotient are defined as follows.

      ( f + g ) ( x ) = f ( x ) + g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) ( f g ) ( x ) = f ( x ) × g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) , g ( x ) 0

Example :

Let f ( x ) = 2 x + 1 and g ( x ) = x 2 4

Find ( f + g ) ( x ) , ( f g ) ( x ) , ( f g ) ( x ) and ( f g ) ( x ) .

( f + g ) ( x ) = f ( x ) + g ( x ) = ( 2 x + 1 ) + ( x 2 4 ) = x 2 + 2 x 3

( f g ) ( x ) = f ( x ) g ( x ) = ( 2 x + 1 ) ( x 2 4 ) = x 2 + 2 x + 5

( f g ) ( x ) = f ( x ) × g ( x ) = ( 2 x + 1 ) ( x 2 4 ) = 2 x 3 + x 2 8 x 4

( f g ) ( x ) = f ( x ) g ( x ) = 2 x + 1 x 2 4 , x ± 2

Another way to combine two functions to create a new function is called the composition of functions .  In the composition of functions we substitute an entire function into another function.

The notation of the function f with g is ( f g ) ( x ) = f ( g ( x ) ) and is read f of g of x .  It means that wherever there is an x in the function  f , it is replaced with the function g ( x ) .  The domain of f g is the set of all x in the domain of g such that g ( x ) is in the domain of f .

Example 1:

Let f ( x ) = x 2 and g ( x ) = x 3 .  Find f ( g ( x ) ) .

f ( g ( x ) ) = f ( x 3 ) = ( x 3 ) 2 = x 2 6 x + 9

Example 2:

Let f ( x ) = 2 x 1 and g ( x ) = x + 2 .  Find f ( g ( x ) ) .

f ( g ( x ) ) = f ( x + 2 ) = 2 ( x + 2 ) 1 = 2 x + 3

Order DOES matter when finding the composition of functions.

Example 3:

Let f ( x ) = 3 x + 1 and g ( x ) = 2 x 3

Find f ( g ( x ) ) and g ( f ( x ) ) .

f ( g ( x ) ) = f ( 2 x 3 ) = 3 ( 2 x 3 ) + 1 = 6 x 8 g ( f ( x ) ) = f ( 3 x + 1 ) = 2 ( 3 x + 1 ) 3 = 6 x 1

Since 6 x 8 2 x 1 , f ( g ( x ) ) g ( f ( x ) ) .