Answer
See graph; asymptote of $y=-1$
domain: $(-\infty,\infty)$ and range: $(-1,\infty)$
Work Step by Step
Step 1. Start from the original function $f(x)=e^x$ shown as the green curve in the figure.
Step 2. To obtain the new curve of $h(x)=e^{x+1}-1$, shift the original horizontally 1 unit to the left; then shift the result vertically 1 unit down to get the red curve as shown.
Step 3. We can identify the new asymptote as $y=-1$
Step 4. We can determine the domain as $(-\infty,\infty)$ and range as $(-1,\infty)$
