Answer
domain: $(-\infty,\infty)$
range: $(-1,\infty)$
asymptote: $ y=-1$
Work Step by Step
The graph of $ f(x)=2^{x}$ is
- always above the x-axis, - the x-axis ( $ y=0$ ) is the asymptote.
- from left to right, always rises and passes through $(-1,\displaystyle \frac{1}{2}),\ (0,1),\ (1,2)$, and $(2,4)$.
Plot these points and join with smooth curve (black, dashed line).
$ h(x)=2^{x+2}-1=f(x+2)-1$, so its graph is obtained by
shifting the graph of $ f(x)$ two units to the left and one unit downward (the asymptote moves down to $ y=-1$ )
Perform the transformation on the four points found above, and join the new points with a smooth curve (red, solid line).
domain: $(-\infty,\infty)$
range: $(-1,\infty)$
asymptote: $ y=-1$
