Composition of Functions

Abstract

1. Present the following explanation to students: Consider the function x x h 4 ) ( . Using an input of 9 to evaluate h, we see that . 6 36 ) 9 ( 4 ) 9 ( h So, h(9) = 6. Since we performed two operations—multiplication and finding the square root—we can think of h as a composite of two functions. Let’s call these two functions f(x) and g(x), with f(x) = 4x and g(x) = x . We can evaluate h(9) by finding the output of f(9) and using that output as the input of function g. First, use 9 as the input for f, and find f(9) = 4(9) = 36. Next, use that output as the input for g and find 6 36 ) 36 ( g . Therefore, 6 ) 36 ( )] 9 ( [ ) 9 ( ) 9 ( g f g f g h  . When we use an output of one function as an input for another function, we are creating a composition of functions. In our example, h is a composition of f and g, which is written as ) ( ) ( or )] ( [ ) ( x f g x h x f g x h  . Both equations are read “h(x) equals g of f of x.”