Answer
Domain: $(-\infty,\infty)$
Range: $(-3,\infty)$
Asymptote: $y=-3$
Work Step by Step
We are given the function:
$$h(x)=e^{-x}-3.$$
Consider the function $f(x)=e^x$ as the parent function.
First we obtain the graph of $g(x)=e^{-x}$ by reflecting the graph of $f$ across the $y$-axis.
Then we vertically shift the graph of $g$ by $3$ units down to obtain the graph of $h(x)=e^{-x}-3$ (see the graph).
From the graph we determine the following elements of function $g$:
- the domain:
$$\text{domain}=(-\infty,\infty)$$
- the range:
$$\text{range}=(-3,\infty)$$
- the horizontal asymptote:
$$y=-3.$$