Difference between f(x) and g(x)
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 )
( fg )( x ) = f ( x ) × g ( x )
Let’s go with an example:
Let’s say f ( x ) = 2 x + 1 and g ( x ) = x 2 – 4.
Find ( f + g )( x ), ( f – g )( x ), ( fg )( x ) and 
.
( f + g )( x ) = f ( x ) + g ( x )
= ( 2x + 1) + ( x2 – 4 )
= x2 + 2x – 3
( f – g )( x ) = f ( x ) – g ( x )
= ( 2x + 1) – ( x2 – 4 )
= – x2 + 2x + 5
( fg )( x ) = f ( x ) × g ( x )
= ( 2x + 1)( x2 – 4 )
= 2×3 + x2 – 8x – 4 _
There is another way to combine two functions to create a new function is called function composition. In function composition we substitute an entire function into another function.
The notation of the function f with g is 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 is the set of all x’s in the domain of g such that g(x) is in the domain of f.
Example 1:
Say f ( x ) = x 2 and g ( x ) = x – 3. Find f ( g ( x )).
Example 2:
Say f ( x ) = 2 x – 1 and g ( x ) = x + 2. Find f ( g ( x )).
Note : The order IF matters when you find the composition of functions.
