An Applet for Visual Composition of Functions

To visually see composition  of functions, h(x)=f(g(x)), it is helps to have three dimensions available.  Then we can make y=g(x) and z=f(y).

The applet has 4 windows, a control panel, an X variable window with a slider to control the value of x, and 2 windows, 2D View and 3D View,  to look at the composition of functions.

2D View has 3 [color coded] graphs, z=f(y) [green], y=g(x) [red], and z=f(g(x)) [blue].

3D View graphs z=f(y) in the y-z-plane, and y=g(x) in the x-y-plane.  [The same green-red-blue color code is used.]  Connected to these curves we have surfaces obtained  by expanding in the unused dimension.
The intersection of the two surfaces traces out a curve for the composite function.

To compare the space curves with their two dimensional counterparts view down the appropriate axis.  [This can either be done with a selection from the view menu of the 3D View window, or by rotating that graph appropriately, or by  selecting 3D View and typing, x to view down the x axis and see the green curve, z to view down the z axes and see the red curve, or Y to view up the y axis and see the blue curve.]

Things to notice:
• To keep everything in sight, the graph is shifted.  in the 3D view, the axis cross through the point (x0, g(x0), f(g(x0))).  In particular this means the linear approximation of f(x) is taken at x=g(x0).
• The scales are the same in all directions.  From the center of a graph to the edge is del when moving along an axis.
• Checkboxes allow you to select wich of the functions f, g, and h=f(g) will be visible.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.