Objectives: In this tutorial, we define the composition of two functions. We look at some examples of compositions of functions. We see an algorithm that illustrates how to get the graph of the composition of two functions from the graphs of the two components. We show how to graph the composition of two functions on a graphing calculator. Finally, we provide some exercises on writing a given function as the composition of two functions. After working through these materials, the student should be able

• to calculate algebraically the composition of two functions;
• to approximate geometrically the graph of the composition of two functions;
• to use a calculator to graph the composition of two functions;
• to express a given function as the composition of two other functions.

Modules:

 Definition. Suppose that f and g are two functions, the composition f g is defined by f g(x) = f(g(x)).

• Discussion [Using Flash]

• Examples.

• A Java applet illustrating the geometrical construction of the composition of two functions.

• Using a graphing calculator to illustrate the composition of functions:

• Drill problems on writing a function as the (nontrivial) composition of two other functions.