## Sunday, July 6, 2014

### Geogebra Series: Inverse Functions

Inverse functions is one of many concepts that can be conceptually enhanced through the dynamic features of Geogebra. Watch the short clip below to see some of the visual aspects that can be incorporated into your presentation.

Inverse Functions Tool

I like this treatment of inverse functions because it starts off with points (solutions) for a given function and we begin by switching the x and y coordinates to find points for the inverse relation.  As the value of "a" changes, students can see the different points of the relation being traced across the screen.

For the first example I picked y=x^2.  I graph the solutions for this relation by letting point A=(a, a^2).   Slider a is set up to go from -10 to 10, so I can graph all of the points for the function y=x^2 for x between -10 and 10.  To plot the points for the inverse relation, I simply switch the x and y-coordinates for point A.  My second point is B=(a^2, a).

Once the two relations are graphed we can observe the reflection over the line y=x.  We can predict points that will be on the graph of y=x^2, and then predict which points will be on the inverse relation.  To do this, simply enter a point into the input bar at the bottom of the screen, and hit enter.

We can discuss what equations will create each graph.  Type into the input bars for Point A and B functions, hit enter, and check the box in order to display each graph.  This is a natural place in the lesson to introduce the procedure for finding inverses (switch x and y and solve for y), because students can immediately check answers when finished by graphing the inverse function.

When students notice that only half of the inverse relation is represented by y=sqrt(x), we talk about why the inverse relation is not a function.  The answer isn't "because it fails the vertical line test", or "because the original function fails the horizontal line test".  The answer is simply that the inverse relation isn't A function because it is TWO functions.  We then add the second function of y=-sqrt(x).  Type into the input bar at the bottom of the screen and hit enter to view this graph.  When we use the vertical and horizontal line tests later as a tool to predict whether one or more functions exist for the inverse relation of a given function, it will be with a greater understanding of why these rules are true.

To try other examples using the Inverse Functions Tool refresh the screen.  Some other examples are:

Point A=(a, a^3), Point B=(a^3,a)
Point A=(a, (a-2)^2-3), Point B=((a-2)^2-3,a)
Point A=(a, 2^a), Point B=(2^a,a)
Point A=(a, sin(a)), Point B=(sin(a),a)

For these examples you might plot the points for A first, make predictions about what the second relation will look like, whether it will be a function, and what the equations will be.  We can then verify these predictions using Geogebra.  For functions such as y=2^x that haven't been studied yet, we can reinforce the understanding of both solution set for an equation as well as inverse relation by choosing points to plot on each graph.

We can come back to this representation of inverse functions in math analysis or precalculus when more function types have been studied.  We can also use this representation to teach specific types of inverse functions such as logarithms or inverse trigonometric functions.

I checked out a section of my Algebra II text on inverse functions, and I found 7 examples.

1. Find an equation of an inverse function.
2. Verify that two functions are inverses of each other.
3. A multi-step word problem
4. Find an inverse and graph both functions.  Restrict the domain.
5. Graph a function, use horizontal line test to decide if inverse exists, then find inverse.
6. Find inverse of a power model (so y=a*x^b)
7. Use the previous inverse power model to make a prediction.

This treatment of inverse functions relies heavily on algebraic manipulation and substitution.  This section is also heavy on vocabulary and mathematical notation.  Having a strong conceptual background grounded in a visual understanding of material can be helpful for these sections.  Being able to access multiple representations of the material by using a tool like Geogebra can also be helpful.