Suppose you have two functions f(x) and g(x). Now you take f(x)/g(x) and want the domain of this new function formed. What you need to do is get the domain of the numerator and the domain of the denominator and find the intersection of the two domains.
f(x) = x^2 - 4
g(x) = sqrt(x^2 - 9)
f(x)/g(x) = (x^2 - 4)/sqrt(x^2 - 9)
domain of numerator is all real numbers.
domain of denominator is (-inf, -3) U (-3, inf)
The intersection is (-inf, -3) U (-3, inf)