#### Answer

$f[g(x)]=g[f(x)]=x$. This means that $f(x)$ and $g(x)$ are inverses of each other.
We know that $x=0$ has to be excluded from both domains because the denominator cannot be $0$ for the function to be undefined.

#### Work Step by Step

We wish to plug $f(x)$ into $g(x)$ to obtain:
$$\displaystyle g[f(x)]=g (\dfrac{1}{x}) \\ =\dfrac{1}{(1/x)} \\=x$$
We wish to plug $g(x)$ into $f(x)$ to obtain:
$$f[(g(x)]=f (\dfrac{1}{x}) \\ =\dfrac{1}{(1/x)}\\=x$$
We see that $f[g(x)]=g[f(x)]=x$. This means that $f(x)$ and $g(x)$ are inverses of each other.
We know that $x=0$ has to be excluded from both domains because the denominator cannot be $0$ for the function to be undefined.