4.4   Composite Functions

Example 1

     y = (x2 - 3)3   is a composite function made up of two functions

                             ~  an inner function   x2 - 3     Let's define this as h(x) = x2 - 3

                             ~  an outer function   x3                  Let's define this as g(x) = x3

   Combining these two functions, we would use the following notation f(x) = g(h(x))  or f ο g(x)    

   (I like to use the first syntax because the second one reminds me of the word - fog.) :-)

    Notice that h(x) is the inner function and g(x) is the outer function.

    To build a composite function, replace all x's of the outer function g(x) with the inner function h(x). 

    The results will be the composite function, f(x).

Example 2

   h(x) = sqrt(1- x^2) is another composite function made up of two functions.

                                      ~ an inner function    1 - x2                  Let's define this as f(x) = 1 - x2

                                     ~  an outer function    sqrt(x)                 Let's define this as g(x) = sqrt(x)

                                                         So, h(x) = g(f(x))