Answer
Domain: $(-\infty,\infty)$
Range: $(-3,\infty)$
Asymptote: $y=-3$
Work Step by Step
We are given the function:
$$h(x)=e^{x+1}-3.$$
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 left.
Then we obtain the graph of $h(x)=e^{x+1}-3$ by vertically shifting the graph of $f_1$ by $3$ units down (see the graph).
From the graph we determine the following elements of function $f$:
- the domain:
$$\text{domain}=(-\infty,\infty)$$
- the range:
$$\text{range}=(-3,\infty)$$
- the horizontal asymptote:
$$y=-3.$$
