Answer
Domain: $(-\infty,\infty)$
Range: $(-\infty,-2)$
Asymptote: $y=-2$
Work Step by Step
We are given the function:
$$g(x)=-e^{x-1}-2.$$
Consider the function $f_0(x)=e^x$ as the parent function.
First we obtain the graph of $f_1(x)=e^{x-1}$ by horizontally shifting the graph of $f_0$ by $1$ unit to the right.
Then we reflect the graph of $f_1$ across the $x$-axis to get the graph of $f_2(x)=-e^{x-1}$.
Finally we obtain the graph of $g(x)=-e^{x-1}-2$ by vertically shifting the graph of $f_2$ by $2$ units down (see the graph).
From the graph we determine the following elements of function $f$:
- the domain:
$$\text{domain}=(-\infty,\infty)$$
- the range:
$$\text{range}=(-\infty,-2)$$
- the horizontal asymptote:
$$y=-2.$$
