Answer
Domain: $(-\infty, \infty)$
Range: $(-\infty, -2)$
Horizontal Asymptote: $y=-2$
Work Step by Step
To graph the following exponential function we will first graph its base (the parent function) and apply a transformations.
As we know the graph of $g(x)=e^x$ is the black dotted graph in the image.
First we apply the following: $g(x)=-e^x$. Which means to reflect the parent graph about $x$-axis. Shown in the image as a green graph.
Then we can get $g(x)=-e^{x-1}$ by shifting the previous graph to the right by 1 unit. So we will get the blue graph in the image.
And finally apply last transformation $g(x)=-e^{x-1}-2$ and graph it by shifting the previous graph downwards by $2$ units. Which is presented by the red graph in the image.
As we have no restrictions domain of the function is $(−\infty,\infty)$.
It has range of $(-\infty, -2)$
And a horizontal asymptote of $y=-2$
