I don't know if this makes function families easier or more complicated, but I realized this week that everything we cover in Secondary Algebra (1 and 2) can be reduced to two basic function families: f(x) = n^x and f(x) = x^n (trig functions excluded, but we don't cover those until precalc at my school). I don't remember this ever getting pointed out to me when I studied functions, and seriously, how many years have I been teaching this? I just find it very interesting.
We made these organizers in class today (and yesterday). We've already got our brains wrapped around transformations and compositions, although we have thus far stealthily avoided talking about operations on functions (adding/ subtracting and multiplying/dividing)... other than to say, "Ugh, two x's, that looks messy." Which of course, it IS, right? And isn't that the point? At least, I think that's my point this time through. Basic functions that involve these families, simple transformations, and compositions where one step follows another in a specified order... this kind of function is not hard to work with. They are logical and orderly. It's only when we start multiplying and dividing or adding/subtracting functions that stuff gets tricky and we need to start pulling out new tricks like factoring and zero product rule, and complex numbers, and extraneous solutions, and reducing rational expressions, and limits, oh my.