Answer
$f(x)$ and $g(x)$ are inverse functions.
Work Step by Step
Let us find the inverse of $f(x)$ by solving for x.
\[ y = \frac{1}{x} \\ \\ xy = 1 \\ \\ x = \frac{1}{y} \]
We see that the inverse function of $f(x)$ is $\frac{1}{x}$ which is also $g(x)$. Therefore, $f(x)$ and $g(x)$ are inverse functions.
Graphically, we see that every point on $f(x)$, when reflected over the line $g(x)$ lies on one point on $g(x)$. Specifically, if a point $(a,b)$ is on $f(x)$, then there is a point $(b,a)$ on $g(x)$.
