It is often easiest to assume "All Real Numbers" and then determine if some subset should be excluded.
I'm not really sure what you are working on. "f(x)= -1/x-3 -1" Is that \(\displaystyle f(x) = \frac{-1}{x-3}-1\)?
The base graph is \(\displaystyle f(x) \ = \ \frac{1}{x}.\)
The (-1) in the numerator reflects it across the x-axis. The (x - 3) in the
denominator verticvally shifts it 3 units to the right. And the subtraction
of 1 after the fraction makes it have a vertical shift of down unit.
The base function has a horizontal asymptote of y = 0, as well as it does not
intersect with its horizontal asymptote (in this particular case.)
When the graph of the base function is relected across the x-axis, the horizontal
asymptote, y = 0, does not change. But when the base function is vertically
shifted 1 unit down, the horizontal asymptote is transformed to y = -1
for the function given. It is still the case that the curve does not cross its
horizontal asymptote (in this case).
For the range, it is:
\(\displaystyle (-\infty, -1) \cup (-1, \infty) \ \ or\)
\(\displaystyle \{y \ | \ y \ne -1\}\)
