Answer
$\text{Domain: }
(-\infty, \infty)
\\\text{Range: }
(-3,\infty)
\\\text{Horizontal Asymptote: }
y=-3
$
Work Step by Step
The black graph is the graph given in Figure 1, $y=
e^x
$.
The red graph is the graph of the given function, $
h(x)=e^{-x}-3
$. It is the result of reflecting the the black graph about the $y-$axis and shifting by $3$ units down.
The red graph has the following characteristics:
\begin{array}{l}\require{cancel}
\text{Domain: }
(-\infty, \infty)
\\\text{Range: }
(-3,\infty)
\\\text{Horizontal Asymptote: }
y=-3
.\end{array}