Answer
a) The functions are inverses of each other.
b) Please see "step by step"
Work Step by Step
If f(x) and g(x) are inverses (of each other, then
a. $f(g(x))=x$ and $g(f(x))=x$ for x in respective domains (of g and f).
(definition, p.337)
b. The graphs of f and g are reflections of each other across the line x=y.
(Theorem 5.6, figure 5.12)
------------------
a.
$f(g(x)=16-[g(x)]^{2}=16-(\sqrt{16-x})^{2}$
$=16-(16-x)=x$
$g(f(x)) = \sqrt{16-f(x)}=\sqrt{16-(16-x^{2})}$
$=\sqrt{x^{2}}=|x|$
This equals $x$ because the domain of $g(f(x))$
is the domain of $f(x),$ which is $x\geq 0$,
for which $|x|=x.$
b. Using a graphing utility, (see below)
we see that the blue graph (f(x) )
and the red graph (g(x)) are reflections of each other about the line x=y:
.
