Answer
$\text{a) The functions are inverses of each other.}$
$\text{b) Please see "step by step" for graph and explanation.}$
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)
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a.
$f(g(x))=\displaystyle \frac{1}{1+g(x)}=\frac{1}{1+\frac{1-x}{x}}$
$=\displaystyle \frac{1}{\frac{x+1-x}{x}}=\frac{1}{\frac{1}{x}}=x$
$g(f(x)) =\displaystyle \frac{1-f(x)}{f(x)}=\frac{1-\frac{1}{1+x}}{\frac{1}{1+x}}$
$=\displaystyle \frac{\frac{(1+x)-1}{1+x}}{\frac{1}{1+x}}=\frac{\frac{x}{1+x}}{\frac{1}{1+x}}=x$
b. Using a graphing device, (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:
.