For my Precalc kids, I started toying with this idea of presenting function operations using diagrams to help kids visually organize domain changes and to see how equations relate to one another.
Here's stab #1 (we started doing this earlier this week. I plan on finishing this tomorrow and easing our way into composition of functions...). My reasoning for organizing it like this is to show kids that addition, subtraction, and multiplication are all very forgiving operations. As long as f(x) is a valid value and g(x) is also a valid value, you can add/subtract/multiply them with no problem. Only division might cause additional exceptions in the domain.
I'm sort of envisioning the kids to then go into something like this, where they can picture composition as one function machine feeding into another one and using that idea to write equations. I also hope that they'll figure out on their own by the end of this second worksheet that g(f(x)) will only be undefined if either f(x) is undefined or if f(x) is some value that will in turn "break" or cause an error in g.
Finally, I'll squeeze in a game/activity, where they get in groups of 3 and split a deck of "cards." Every round I'll call out some order of composition -- for example, g(f(h(x))) -- and the kids would need to write the resulting equation and find the domain restrictions on that formula. And then, with those same 3 cards I'd reverse the composition order, and they'd do it again. It's not terribly fun of a "game", I guess, but as far as activities go, I hope it will be a little better than doing straight problems on paper, because they'd hopefully notice that the domain restrictions only come into play for certain types of operations (ie. square root), regardless whether that operation happens first or last in the composition.
Thoughts or suggestions?
...By the way, I recently read on someone else's teaching blog that they have all these really wonderful things planned for their Precalc kids. It made me feel a little math-envy, but alas, my kids need all of the basic reinforcements that they can get. (They're coming along, and are actually understanding words that are written on paper with numbers inserted in between!!!! It seems like half a miracle considering the zombie-esque state that they were in when I got them back in August.) But, our sloooowness in progress is making me very worried about their future, and I'm going to try my best not to become a total stressball over this during the next few months. It's going to take me another couple of weeks to just get through all of the Algebra 2-ish review-ish stuff with them, and then whatever trig I can squeeze in to the rest of the year, I'll have to be happy with!! The end of the year is coming SO FAST.