## Intermediate Algebra (12th Edition)

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