There are many significant ideas about inverse functions which can be more effectively taught using technology as a motivation and tool for constructivist learning. As a premise for this discussion, I offer a summary by Erik De Corte:

"There is now a substantial body of research in instructional psychology showing that learning is an active and constructive process. Learners are not passive receptacles of information, but they actively construct their knowledge and skills through interaction with the environment..."

From that perspective, I would begin this lesson by asking students to explore the composition of functions. Without giving any preliminary explanations, I would use a spreadsheet application to show a few examples of two operations applied in sequence to a set of numbers resulting in the orginal input. I would discuss the event as some strange phenomena shrouded in mystery. Then I would invite students to make a few conjectures on what is happening, and how they might find a second operation that would result in the same output. If students are in a computer lab setting, they can quickly test predictions of trying to find the "second operation" which will give the intended results.

(If a spreadsheet application is not available, this same experiment can be done on graphing calculators with list capabilities. Let L2 be defined by a function in the "Y=" screen with L1 as input. Similarly, let L3 be defined with L2 as input.)

As students make and test their conjectures about the nature of the "second operation," have them share their failures and their successes, recording all observations. Simultaneously, students should be discussing the merits and ideas of different conjectures, and through discussion and experimentation, develop increasingly more sophisticated ideas about what is happening and how to find an operation which "undoes" the first one. This activity depends upon the ability to quickly generate the data to test one's hypothesis. Without technology, a teacher could begin with one small example and ask for conjectures, but the "test" would be limited to the teacher's responses or a very few calculations in the brief time allowed. This form of interaction rewards quick, "right" answers and diminishes the analytic initiative in all but the quickest of students.

This process can be extended to practicing finding the inverse of a function. I would begin the drilling with the exercise: "Using the spreadsheet to test your ideas, develop an algorithm for finding the inverse of any given function." What is most exciting is that students will most likely generate different algorithms. The class can then be given the task of evaluating various solutions based on their accuracy and efficiency. For extra practice, such evaluation could be done by checking multiple problems by hand. This same assignment could be given without using a spreadsheet, but it would be far more daunting to most students. I suspect there is the potential for "tricking" students into thinking when they believe the computer is doing all the work.

Another important topic of inverse functions is the relationship between the domain and range. Again, the essential difference between a technological or a non-technological approach is the amount of demonstration or testing that is logistically feasible, and therefore how much material must be delivered by the teacher due to lack of time. But there are also some interesting situations that can arise. Graphing calculators are not designed to draw relations (unless as data sets), only functions. Give students a parabola, ask them to draw its inverse, and watch them draw sideways "U"s. Then ask them to check it with the calculator, and wait for someone to shout, "Hey! Why did it only draw half the graph?" How much more will they be interested in finding this out than if I had merely explained the correct process?!

The spreadsheet can also be effectively used to reveal the domain of an inverse function. Students sometimes become too familiar with graphs and may not notice that the graph of a standard parabola has only positive points with respect to the y-axis. In a spreadsheet listing of the results, however, you can ask a class to describe the type of numbers in a given column. This visual, side-by-side, format is also good for demonstrating how the range of the function becomes the domain of the inverse, but not necessarily vice-versa. Most spreadsheets also come with graphing capabilities which allow you to see the numerical and spatial representations side-by-side. (Graphing calculators can also show both representations, though not usually simultaneously.) A certain amount of independence in learning is provided by these technologies which increases the potential for student ownership of knowledge. Without technology, students tend to be more dependent upon the teacher or textbook as a dispenser of knowledge.

Bibliography:

DeCorte, Erik. "Technology-supported learning in a stage of transition: the case of mathematics," Lessons from Learning. R. Lewis and P. Mendelsohn (eds.) Elsevier Science B.V. (North-Holland), IFIP, 1994.