Answer
The graph of $y=3^x+2$ involves a $2$-unit shift upward of the parent function $f(x)=3^x$.
The given function has:
domain: $(\infty, +\infty)$
range: $(2, +\infty)$
horizontal asymptote: $y=2$
Work Step by Step
Recall:
The graph of the function $y=f(x)+k$ involves a vertical shift ($k$ units upward when $k \gt0$, $|k|$ units downward when $k\lt0$) of the parent function $y=f(x)$.
The given function has $f(x)=3^x$ as its parent function and can be written as $y=f(x)+2$.
With $k=2$, the graph of the given function involves a $2$-unit shift upward of the parent function $f(x)=3^x$.
Recall:
The function is $y=a^x$ has:
(1) domain: $(-\infty, +\infty)$;
(2) range: $y\gt 0$; and
(3) horizontal asymptote: $y=0$
With a vertical shift of the parent function $f(x)=3^x$ of $2$ units upward, the function $y=3^x+2$ has:
(1) domain: $(-\infty, +\infty)$;
(2) range: $y\gt 2$; and
(3) horizontal asymptote: $y=2$
