I'm teaching about absolute value functions these days, and about rewriting absolute value functions as piecewise functions. The students have a lot of trouble with this. In the hope that I'll remember to check out my notes on what they're struggling with before next time I teach this stuff, I'm storing them here.

The big problem seems to be to understand that the restricted domain is just that, a specification of what interval of the independent variable we are concerning ourselves with, and that the domain is not somehow "a solution" or part of the actual function or something. I am clearly unclear about just what is the issue, and how to go about filling in the missing parts. Apparently Dan Greene has figured out ways of approaching students' tendency to "flip back and forth between x and f(x) in their minds". I guess I should backtrack and spend a whole lesson or two on picking out the segment of a graph corresponding to a particular x-interval. Not that there's really time for that, but... it is needed.

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