Answer: Yes it's possible to have negative numbers in the domain of a rational function.
Consider the example:
f(x) = (3x^2+5)/(9x^3-27x+10)
Here we can plug in something like x = -7 and get some y output without worrying about a division by zero error. So a rational function can have negative numbers in the domain. The only thing you need to watch out for is when the denominator is zero. Dividing by zero is not allowed.
For instance, let's say we had
f(x) = (x+5)/(x-12)
Here we cannot plug in x = 12 into the function because that would make the denominator x-12 turn into 12-12 = 0. So to find any restricting x values, you would need to set the denominator equal to zero and solve for x.