Answer
See below.
Work Step by Step
Step 1. To obtain the graph of $f(x)=5-e^{-x}$ from $y=e^{x}$, reflect the curve across the y-axis, reflect across the x-axis, and then shift 5 units up. See graph.
Step 2. We can determine the domain is $(-\infty,\infty)$; the range is $(-\infty,5)$; and the horizontal asymptote is $y=5$.