<s>[INST] Does dividing a common factor out from numerator and denominator of a rational function create a new function with different domain? Suppose a function is defined as $  f(x) =\frac{x^2 - 9}{x-3}. $
If we divide the common factor $ x-3 $ from both the numerator and denominator :-
$$ \frac{x^2 - 9}{x - 3} \\ = \frac{(x+3)(x-3)}{x-3} \\ = x+3 $$
1) Is $ x+3 $ a different function from the original $ \frac{x^2 - 9}{x - 3} $ ?
2) Since $ \frac{x^2 - 9}{x - 3} $ does not have 3 in its domain, when we simplify it to become $x + 3$, does it now have 3 in its domain ?
 [/INST] $x+3$ is different from $\frac{x^2-9}{x-3}$, but only in the domain. All the simplification does is to add $x=3$ to the domain.
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