Today at the end of Precalculus today I asked if any kids had any questions/topics they wanted a quick review on for an assessment we’re having tomorrow. (We lost a week due to Hurricane Sandy, so it’s been a while since they’ve worked on some of the topics.) One of the topics was inverse functions, so I gave a quick 3 minute lecture on them, and then we solved a simple “Here is a function. Find the inverse function.” question. They then wanted an example of a more challenging one, so I made up a function:
And then we went through and solved for y. And we found that …
… the function is it’s own inverse. Yup, when you go through and solve it, you’ll find that is true. (Do it!)
I said: “WAIT! Don’t think this always happens! This is just random! Really! This is random!”
But I just had the thought that this might be good to capitalize on. So I sent my kids the following email:
I don’t know if anyone will bite, but I hope that someone decides to take me up on it. I already have a few ideas on how to have them explore this! (Namely, first exploring 
We’ll see… I’m trying to capitalize on something random from class. I hope it pans out.
Continuous Everywhere but Differentiable Nowhere 
Omg, Sam, you are my hero. This happened to me in Pre-Calc this year too with a very similar function. My first thought was, “It can’t be. Re-check the algebra.” Not a lot of algebra to check. Run to calculator. Discover the amazingness. I was stoked.
You’ve probably already thought of this, but one good exploration for the kids (especially after they’ve done some algebra on the functions) would be to graph the functions you gave using sliders for a, b, c, and d. Fun!
Also, abs value functions are sometimes their own inverse, depending on how you restrict the domain. Not quite as challenging as rational functions, but I think it provides a good discussion nonetheless and would be a very nice example to include if someone does bite and decides to write a paper on it.
Please keep us posted on this!!
Probably you know this already, but rational functions like the one you found are, in the complex plane, very interesting transformations! Look here: http://en.wikipedia.org/wiki/Möbius_transformation
I remember coming across this when I taught pre-Calc and it’s pretty interesting. I was thinking about why this happens and noticed that it’s whenever the horizontal and vertical asymptotes are the same. So with your (ax-b)/(cx-d) the horizontal asymptote is a/c and the vertical asymptote is d/c. They are equal when a=d. As long as a=d this rational function will be it’s own inverse. Thanks for sharing! I hope your kids figure this out. I bet some will.
Hey Sam,
Long time listener, first time caller. Love the Math Journal idea, and everything else you do.
Now a somewhat off topic question, but since you mentioned “review” in this post, I guess it fits. Reading the blog, it seems like review isn’t something that you do formally before you give an assessment, am I right in thinking that? I’m introducing SBG into my school’s math classes for the first time, and haven’t been giving formal reviews sheets prior to assessments since the expectation for my students is that they should always be reviewing a little bit every night. However, I think the power of the Review Sheet has caused them to be lazy brained and a lot of them are doing poorly on their assessments due to their lack of review on their own. Should I go back to holding their hand and give them review sheets or just doing it like I have been? (To be clear, I give them the skill that they will be assessed on, the section it comes from in their notes, the homework problems that they’ve done on it, and other problems in their books to try for practice many days before the assessment.)
Let me know what you think
-Eric
Hihi-
I don’t review in calculus where I do SBG. However, I am teaching precalculus this year for the first time, and I’m not doing SBG in that class. (To do SBG, I need to get the other teacher on board, and also I don’t feel comfortable switching to SBG until I am comfortable with the course so I can make standards that make sense to me.) Thus, my precalculus class is more traditional.
As for the review sheet, you give them so much, so I don’t think you need to give them a review sheet: “To be clear, I give them the skill that they will be assessed on, the section it comes from in their notes, the homework problems that they’ve done on it, and other problems in their books to try for practice many days before the assessment.”
Best,
Sam
One of the questions on my pre-calc test (and it’s a concept important enough I announce beforehand what it will be) is “What happens geometrically when taking the inverse of a graph?”
A natural extension is “what happens geometrically when the inverse of a graph is the same as the original graph?”
For your question, I’d say it is easiest to answer in reverse — figure out how to construct rational functions that have the line y=x as a line of symmetry.