Answer
domain $(-\infty,\infty)$, range $(1,\infty)$
See graph.
Work Step by Step
To graph function $f(x)=2^{x+3}+1$, start from $y=2^x$, shift the curve 3 units to the left, then shift up 1 unit.
For $f(x)$, we have domain $(-\infty,\infty)$, range $(1,\infty)$
See graph.