I put the following problem on a recent homework assignment:If you can’t read it, it gives a table that maps the number of years Han Solo and Chewbacca* have been smuggling to the total value of goods they smuggled that year. I ask whether the inverse function exists, what it means in that context, and how you could change the table so that it doesn’t exist. Then, remembering that meaning matters, I ask the students to come up with a reason why someone (Han, Jabba, an Imperial customs officer…) might care about the inverse function. I was thinking something like, “If they know they made a certain amount at some point, but couldn’t remember which year…” with suitable dressing about why Jabba the Hutt is after them for not giving him the right cut or something.
The answers students gave me were baffling. There were, of course, the silly ones like “If a strong enough ion blast were to overload the Death Star’s main cannon as it fired, it could invert the world due to….” Well, you get the idea. But over and over again, students told me that Han would care about this function because he would want to see that he was making more money, that the amount that he smuggled was increasing, etc.
Now, it’s a lot easier to see that the smuggled amount increases as time goes on if the input is time and the output is the amount smuggled. That’s the whole point of talking about increasing functions. It’s a lot harder to tell if you’re looking at the inverse function, although it could certainly be done.
Finally, I found a student who wrote this:
And a lightbulb went off.
They weren’t telling me why the function was useful, but why the existence of the function was useful. And when I looked at my questions again, I realized why. The previous two questions had been about the existence of the function. That’s what I’d put on their minds. Of course they were going to tell me about the existence of the function!
I’m not sure how to change this problem to get what I want. On the other hand, I’m not sure that I should change the problem to get what I had in mind. Their interpretation of the question gives a question worth thinking about. The goal was to make inverse functions meaningful. If this makes it meaningful…ok. Maybe a good follow-up would be, “Is the inverse function the easiest way to figure that out?”
Still thinking about this…
*One class realized that I misspelled Chewbacca on the key. This lead to a student asking me how you spell his nickname: Chewie or Chewee.
Me: “It’s Chewie.”
Him: “Are you sure?”
Me: “I spent my entire childhood reading Star Wars novels. I’ll stake my life on this.”
Different student: “I think Mrs. L might be a Star Wars fan…”
Yes, dear child, I am.