## Intermediate Algebra (12th Edition)

$y \text{ is NOT a function of }x \\\text{Domain: } (-\infty,\infty)$
$\bf{\text{Solution Outline:}}$ To determine if the given relation, $x-y\lt3 ,$ is a function, solve first for $y$ in terms of $x.$ 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 inequality, the given relation is equivalent to \begin{array}{l}\require{cancel} x-y\lt3 \\\\ -y\lt-x+3 \\\\ -1(-y)\lt-1(-x+3) \\\\ y\gt x-3 .\end{array} If $x=0,$ then $y\gt-3.$ That is, the ordered pairs, $\{ (0,-2),(0,-1) \}$ satisfy the given equation. This means that the value of $x$ is not unique. Hence, $y$ is NOT a function of $x.$ The variable $x$ can be replaced with any number. Hence, the domain is the set of all real numbers. The given relation has the following characteristics: \begin{array}{l}\require{cancel} y \text{ is NOT a function of }x \\\text{Domain: } (-\infty,\infty) .\end{array}