Answer
Domain: $(-\infty, \infty)$
Range: $(-3, \infty)$
Horizontal Asymptote: $y=-3$
Work Step by Step
To graph the following exponential function we will first graph its base (the parent function) and apply a transformation.
As we know the graph of $h(x)=e^x$ is the black dotted graph in the image.
Then we can get $h(x)=e^{x+1}$ by shifting the parent function to the left by 1 unit. So we will get the blue graph in the image.
And finally apply last transformation $h(x)=e^{x+1}-3$ and graph it by shifting the previous graph downwards by $3$ 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 $(-3,\infty)$
And a horizontal asymptote of $y=-3$