Exploration: Inverse Functions
1. In the applet below, points C, D, and I are movable by dragging them around with your mouse. The line y = x (the identity function) is not movable. Begin this activity first by dragging the points around.
Points E, F, and G are also movable. Consider the identity function as a line about which we can reflect C, D, and I. Drag points E, F, and G to the places where C, D, and I would reflect. What would their coordinates be?
2. Now show the actual reflections, C’, D’, and I’ by clicking on the boxes to the right. Were your approximations accurate?
3. Now hide E, F, and G (click on their boxes). Play with the visual of the point and its reflection by dragging points C, D, and I around. Try to drag any of the reflection points. Can you move them? Why or why not?
4. Now, find the equations to the lines that pass through the following pairs of points:
a. C and I
b. C and D
c. C’ and I’
d. C’ and D’
Please write any observations you have about the relationships between the segments and their corresponding reflections.
5. Now click the boxes to show CI, CD, C’I’, and C’D’. Write the piecewise defined function for each line (except for the identity function).
Write any observations you may have.
6. Consider other curves. Imagine a parabola passing through C, D, and I. How would that shape reflect about the identity function? What about a cubic function? How would that reflect? If you were to give someone directions on what to do to make an exact representation of the reflection of any function about the line y = x, what would you say?