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)
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
obtained by expanding in the unused dimension.
The intersection of the two surfaces traces out a curve for the
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
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.