Function composition: The next topic that we have to discuss here is that of function composition. The composition of f(x) & g(x) is

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

In other terms, compositions are evaluated by plugging the second function listed in the first function listed.  Note that order is significant here.  Usually interchanging the order will result in a different answer.

Example   Given  f ( x ) = 3x² - x = 10  and  g ( x ) = 1 - 20 x determine each of the following.

(a) (f o g ) (5)

(b) ( f o g ) ( x )

Solution

(a) (f o g) (5)

In this case we've got a number rather than an x but it works in precisely the same way.

( f o g ) (5) = f (g (5))

= f ( -99) = 29512

 (b) ( f o g ) ( x )

( f o g ) ( x )  = f ( g ( x ))

= f (1 - 20x )

= 3(1 - 20x )²  - (1 - 20 x ) + 10

= 3(1 - 40x + 400 x² ) -1 + 20 x + 10

= 1200 x² -100 x + 12

Compare this answer to the next part & notice that answers are NOT the similar.  The order wherein the functions are listed is important!