Answer
$ a.\quad$The graph is symmetric about the origin.
$b.\quad f(x)=f^{-1}(x)$
Work Step by Step
$a.$
See graph.
The graph is symmetric about the origin.
$b.$
Solve the equation for x, writing $y=f(x)$
$ y=\displaystyle \frac{1}{x} \qquad$... raise both sides to $(...)^{-1}$
$\displaystyle \frac{1}{y}=x\qquad$... interchange $x\leftrightarrow y$
$y=f^{-1}(x)=\displaystyle \frac{1}{x}=f(x)$
