It always amazes me how my students get all the way to precalculus with such an underdeveloped sense of function. I often hear teacher’s complain about how their students can’t deal with fractions, or solve equations, but I find it much more alarming that they don’t have a basic understanding of domain, range, and function.
Sure, they can rattle off some basic facts about domain being x’s and range being y’s, but how much do they really understand about the role they play? Do you want to find out? Are you sure? If you do, try this:
Hand your students two graphs on the same coordinate plane, one f and one g. They can be pretty basic. It is best if they have some easily identifiable domain and range.
As them to identify the domain and range of both graphs (this is pretty easy.) Now, ask them to evaluate something like f(2) using the graph. Many of them wont’ know what that means. They will ask for the equation. It get’s better. Ask them where f(x) > 0. Ask them where f(x)=0 and what f(0) are. Ask them what (f + g)(2) is. Ask them what the domain of (f+g)(x) is.
For the clincher, ask them what the domain of f(g(x)) is. Don’t give them the equations. Make them figure it out from the graph. Start talking to them about getting the range of the inner function to fit in the domain of the outer function, and watch their eyes glaze over.
Last of all, when you really want to stump even your best kids, ask them where f(g(x))=0. Whoa.
These kidns of basic questions, which involve no algebra at all, will get your kids (and you) thinking more than you ever have about domain, range, and function.