Writing Inverse Functions

I like Sam Shah’s approach to teaching function inversion. It’s pretty much what I had intended to do this year, but somehow forgot about at some point. Hopefully writing these notes here will increase the odds that I get it right next time around. The original inspiration was Mr. K’s approach to solving two-step equations, and the following will make no sense without having read that entry, so do that first.

I showed my Algebra 2 students Mr. K’s series of boxes and arrows for solving equations early in the year, and they thought it was fun. We did more complicated cases such as 5-3(x+5)=2 and -2-(2x+3)=-5. It was great for reviewing Order of Operations in a novel way. Many students had started out subtracting 3 from 5 in the fist equation, and in the second case very few students were able to identify “multiply by -1” as one of the operations in the sequence. However, after working through a few cases they got quite fluent at writing sequences of operations such as for the equation -2-(2x+3)=-5, where the operations on x are:

1. Multiply by 2
2. Add 3
3. Multiply by -1
4. Add -2

The inverse operations, in the reverse order, are

1. Subtract -2
2. Divide by -1
3. Subtract 3
4. Divide by 2
   

Applying these operations to -5 solves the equation above. Applying the same operations to a variable yields the inverse of the function y=-2-(2x+3)

My plan was to bring this visual in again, as a recurring Opener problem perhaps, with every new operation we covered. The processes of squaring and taking a square root, of exponentiating and taking the logarithm, could presumably be organized nicely with the same kind of diagram that Mr. K used to organize the process of undoing linear operations on x. In Algebra 2, the reassuring string of little boxes could make for a beautifully transparent structure for inverting a function, with built-in checks at every step. Practicing with numerical input and output values first would likely be helpful.

Maybe I'll try this when reviewing just to see if it works.