Composition of Functions

Below, the purple point traces out the composition: g(f(x)).

Walk your way back and forth through it, and see why the result IS the composition.

The light blue diagonal line is the line y = x, which just simplifies the diagram a bit, since reflection off of it is the same as converting the output from f(x) into the input for g(x).

1. Find the values: g(f(-3)), g(f(0)), g(f(4))

2. Over on the left, you'll see the definitions for f(x) and g(x), and down in gray, h2 is the composition g(f(x)) written out. If you want to, you can try changing the functions f(x) and g(x) to observe the composition. If you right click on f(x) and redefine it, try letting f(x) = some bounded function like sin(x). Then return f(x) to what it was, and try replacing g(x) with sin(x). As you consider the results, you're looking into some rather deep mathematical ideas involving the theory of functions and dynamical systems. For more interesting results, try redefining h2(x) = f(g(x)) and h2(x) = f(f(x))

Dave Matthews, January 16, 2008, Created with GeoGebra