So I put all the big information into a pretty bare-bones worksheet, and threw ‘em a few curveballs. No one noticed that the inverse of x^2 only spits out half the original’s graph, so we’ll have to talk about that tomorrow, but they were getting pretty angry that I wouldn’t just flat-out tell them how to find the inverse of 1/x, which is always fun.

Again, it’s pretty bare-bones, but it beat the hell out of lecturing this stuff, which can be a slog:
https://docs.google.com/open?id=0B1klX05TDTbQN2VhYjQ0YzItOGM0MC00NGVlLTgxYzgtOWI5ZDQyNDVjNzBi

This approach totally blew up how I teach this, which I’m actually doing this week.

I actually included the step-by-step idea with composition of functions, and today I introduced y = x as the identity function, whose steps only include “do nothing.”

When doing the step listing idea this year, I threw out the usual “think of a function as a machine, think of compositions as stringing machines together.” Halfway through class a student asked to come to the board with their own explanation, and showed the class the machine thing.

I’m also augmenting this approach by having the students do rules and inverses as listed steps first.

Thanks so much for this.

Thanks for your comment. We definitely talk about 1 to 1 functions — and my students know that they can’t find f^{-1} if f(x)=x^2. I harp on this point a lot!

I just wanted to highlight what’s going on for simple 1 to 1 functions.

Best,
Sam

Now “f^{-1}(x)” is defined as a multi-valued mapping.

f^{-1}(x)={y:f(y)=x}

Indeed f^{-1}(0)=k\pi. So let x=0
f(x)=0 and f^{-1}(f(x))=k\pi; NOT 0.

Now f(f^{-1}(0))=0.

f(f^{-1}(x)=x but f^{-1}(f(x)) not necessarily x.

A simpler example, f(x)=x^2:

f(f^{-1}(1)=1 but f^{-1}(f(x)) plus or minus 1

i call “invert each step and
reverse the order” the
“shoes and socks” theorem.
i forget where i learned to do so.
it appears to be well-known.

.

We then talk about all the other function operations as manipulations of these machines. Composing functions is rigging them so that the chute from one goes right into the slot of another. Inverse functions are hitting the “reverse” button on them. So, with those concepts in mind, we have two duplicate machines connected to one another where the chutes (or slots, depending on which order you’re doing) are glued in the middle and one of them runs in reverse.

For something like your example, we might break down the original function into composite functions and draw the machine connection diagram.

I think the visualization hits home with some of the kids. I like your analytic view, too, though. The pattern is easily seen with the words/steps.