This has bugged me for quite some time. In algebra (and more advanced) classes, it can be… well, not common, but not rare to have an equation such as y=x^2/x. In many cases, we’re encouraged to cancel and get y=x.

This… bothers me, because it seems to me you’re implicitly changing your function’s domain from (-inf,inf) - {0} to (-inf,inf). Now, this didn’t go completely unmentioned way back in pre-calc, we definitely talked about asymptotes and holes in graphs, but it was always in the context of “when graphing x^2/x… but you should generally just cancel the terms, and then there will be no hole!”

I never understood why you’re allowed to transform (x^n)/(x^m) **where m<=n** (you don’t get this problem when m>n) to x^(n-m)*; it seems like implicitly changing the domain of your function should be problematic, or at least considered. Why do we say that these two statements are equivalent when you have to invoke a domain change to do it? Is it just that it usually doesn’t matter? Just bad teaching?

- Qualification: Obviously x^(n-m) = (x^n)/(x^m) where x is in (-inf,inf)-{0} for both expressions, but the we were instructed to do it without fail implicitly changed the domain of x^(n-m) to also include zero. In other words f(0) shouldn’t have worked for either, but we always treated it as if the x^(n-m) expression magically could handle f(0) now.