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When I am given a rational function, say,

f(x) = (x

^{2}+x-6)/(x+3)

say I want to look at the domain of this function. Well the first thing that catches my eye is the fact that if I plug in x=-3, the denominator will be 0, which would be undefined. So I immediately think, the domain of this function is all real numbers except -3. But then when I factor the numerator, I end up realizing that f can be written as:

f(x) = (x-2)(x+3)/(x+3)

I think to myself, ok, I can just cancel the (x+3) from the denominator and numerator, and now I just have the function f(x)=x-2

Great! But that's not the case. I mean at least my calculus book gives me the impression that its not. In my book when introducing limits, they give that f(x) I listed as an example of a function where the function is not defined at a point because of a zero in the denominator. But why can't I just factor this out and cancel the (x+3) from the denominator? It makes me feel like whenever I'm doing ANY calculation with rational functions I shouldn't cancel out any of these (x+c) terms that are common in the numerator and denominator because I feel I'll be losing important information about the function (it's domain)! In other words: I no longer feel I can say (x+3)(x-2)/(x+3) = (x-2) because the function on the left doesn't have the same domain as the function on the right, so they aren't the same function!

I just don't understand.