After years of using a trip directions metaphor to describe what an inverse function is, I decided to develop this year’s lesson around that metaphor. I started the lesson with a review of function operations and composition on a series of tables, but curiously put a question on there with some notation that didn’t look familiar. The question was an inverse function question. After showing the class the answer, they decided they needed more information about how to “do the problem.” You know that’s not good enough. They know that’s not good enough. Yet, they still asked. And here’s how the lesson on inverses progressed…
I opened with this homemade video:
Then, a student gave directions on how I went from school to Publix:
Thus began our discussion of a function inverse. I asked another student to get us back from Publix to school (following the same route):
This is where we defined a function inverse. If a function maps you from an input to an output, its inverse maps you from the output back to the input. That’s a one-line interpretation of the conversation; a lot more went into it. But you get the gist.
The next step was to figure out how to calculate the inverse function. The first thought a few students had was to just turn left where we turned right and turn right where we turned left. So we compared:
The class decided that we had to do the opposite operation (turn) but it had to be done in the opposite order. The last turn into Publix is the first turn you undo when leaving Publix.
The rest of the lesson was rather pedestrian with some examples and practice. I’m hoping that this understanding of the meaning of an inverse of a function pays off.
Cloud Math 
Thought-provoking post. Combines the thought processes of Locke and Rousseau with the mathematical wisdom of Muḥammad ibn Mūsā al-Khwārizmī. Of course, the question it leaves hanging in the air, like an over zealous flapper doing the Charleston atop an aircraft, is whether or not Publix is even a true being. Perhaps, in our search for directions to this “Publix” entity, we actually search for the directions to our own state of being; our consciousness, if you will. I’d like to close my comment with a quote from the great Euclid:
The laws of nature are but the mathematical thoughts of God.
Please take note that I include this quotation not because it is at all relevant to my other comments, but because it was literally the first quote that came up when I typed in “Euclid quote.”
Thank You.