Answer
For the graphs of these functions, see the image below.
(a) $f(x)=2^{-x}+4$
Domain : $(-\infty, \infty)$
Range : $(4, \infty)$
Asymptote : It has horizontal asymptote $y=4$
(b) $g(x)=\log_3(x+3)$
Domain : $(-3, \infty)$
Range : $(-\infty, \infty)$
Asymptote : It has vertical asymptote $x=-3$
Work Step by Step
(a) $f(x)=2^{-x}+4$
We have no restrictions, so the domain is: $(-\infty, \infty)$
The first part of this function ($2^{-x}$) is always more than $0$. For infinitely large values it gets infinitely small non-negative value (it approaches $0$), For infinitely small (negative) values it gets infinitely large value, so the range of this function is: $(4, \infty)$
As mentioned earlier the value of the function approaches but never gets less than $4$, so It has horizontal asymptote $y=4$
The graph of this function has no $x$-intercept. For $y$-intercept see the image above.
(b) $g(x)=\log_3(x+3)$
Due to the definition of logarithms $x+3\gt0$ => $x\gt-3$, so the domain is: $(-3, \infty)$
It has range of: $(-\infty, \infty)$
Asymptote : It has vertical asymptote $x=-3$, simply because the argument of a logarithm can't be equal or less than $0$, so it approaches but never crosses the line.
