The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
GeoGebra, Markus Hohenwarter and Judith Preiner

# Composite Functions

Apart from geometrical objects GeoGebra provides functions in x as separate type of objects. These functions can be combined seamlessly with geometrical objects in one construction. Let's examine the new possibilities of this feature by composing two functions g(x) = sin(3 x) and h(x) = x2 / 4 to get f(x) = g(h(x)). In the following dynamic construction we show how this composition can be constructed geometrically.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)

Editor's note, May 2014: For an HTML5 version of the above applet, click here.

Point D has the x-coordinate x0 while its y-coordinate is f(x0). By dragging point x0 with the mouse you can examine the course of the composite function f(x) = g(h(x)) which is created as the trace of point D (see how this construction was done). With the dynamic construction above you are able to analyze the graph of the composite function f(x). You can investigate either the trace or the locus line of point D.

• To turn off the trace, right click (Mac OS: apple click) on point D and uncheck "Trace on" in the context menu.
• To create the locus line you activate the tool "Locus" in the toolbar and click successively on point D and point x0.
• To check whether our construction was really right, you could enter `f(x)= g(h(x))` into the input field to discover that its graph matches the locus line of point D perfectly.

A special feature of GeoGebra is that you can drag the graphs of the functions g and h with the mouse, whereby the graph of the composite function f changes dynamically. Furthermore, you can also modify the equations of the functions g and h directly in the algebra window. Thus, this single construction allows you to explore virtually every composition of two functions. Note, that you can use the tools "Move drawing pad" as well as "Zoom in" and "Zoom out" to make important details of the construction visible in the graphics window of the mathlet.