Intermediate Algebra (12th Edition)

$y \text{ is a function of }x \\\text{Domain: } \left[ -\dfrac{7}{4},\infty \right) \\\text{NOT a linear function}$
$\bf{\text{Solution Outline:}}$ To determine if the given equation, $y=\sqrt{4x+7} ,$ 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 expression $\sqrt{4x+7}$ will produce only $1$ value of $y.$ Hence, $y$ is a function of $x.$ The radicand of a radical with an even index cannot be negative. Hence, \begin{array}{l}\require{cancel} 4x+7\ge0 \\\\ 4x\ge-7 \\\\ x\ge-\dfrac{7}{4} .\end{array} Hence, the domain is the set of all numbers greater than or equal to $-\dfrac{7}{4}.$ 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[ -\dfrac{7}{4},\infty \right) \\\text{NOT a linear function} .\end{array}