Answer
Domain: $(-\infty, \infty)$
Range: $(4, \infty)$
Horizontal Asymptote: $y=4$
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 $f(x)=e^x$ is the black dotted graph in the image.
Then we can get $f(x)=e^{x−3}$ by shifting the parent function to the right by 2 units. So we will get the blue graph in the image.
And finally apply last transformation $f(x)=e^{x-3}+4$ and graph it by shifting the previous graph upwards 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 $(4,\infty)$
And a horizontal asymptote of $y=4$