## Intermediate Algebra (12th Edition)

$y \text{ is a function of }x \\\text{Domain: } \left( -\infty,6 \right)\cup (6,\infty) \\\text{NOT a linear function}$
$\bf{\text{Solution Outline:}}$ To determine if the given equation, $y=\dfrac{7}{x-6} ,$ is a function, check if $x$ is unique for every value of $y.$ To find the domain, find the set of all possible values of $x.$ A linear function is any equation that can be expressed as $f(x)=mx+b.$ $\bf{\text{Solution Details:}}$ For any $x,$ the quotient of $7$ and $x-6$ will produce only $1$ value of $y.$ Hence, $y$ is a function of $x.$ The denominator cannot be zero. Hence, \begin{array}{l}\require{cancel} x-6\ne0 \\\\ x\ne6 .\end{array} The domain is the set of all real numbers except $6.$ Since the given equation cannot be expressed as $f(x)=mx+b,$ then it is not a linear function. The given equation has the following characteristics: \begin{array}{l}\require{cancel} y \text{ is a function of }x \\\text{Domain: } \left( -\infty,6 \right)\cup (6,\infty) \\\text{NOT a linear function} .\end{array}