#### Answer

$y \text{ is a function of }x
\\\text{Domain: }
\left( -\infty,0 \right)\cup\left( 0, \infty \right)$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To determine if the given equation, $
xy=1
,$ is a function, isolate first $y.$ Then check if $x$ is unique for every value of $y.$
To find the domain, find the set of all possible values of $x.$
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
xy=1
\\\\
\dfrac{xy}{x}=\dfrac{1}{x}
\\\\
y=\dfrac{1}{x}
.\end{array}
For each value of $x,$ dividing $1$ and $x$ will produce a single value of $y.$ Hence, $y$ is a function of $x.$
The denominator cannot be $0.$ Hence,
\begin{array}{l}\require{cancel}
x\ne0
.\end{array}
The given equation has the following characteristics:
\begin{array}{l}\require{cancel}
y \text{ is a function of }x
\\\text{Domain: }
\left( -\infty,0 \right)\cup\left( 0, \infty \right)
.\end{array}