#### Answer

$y \text{ is a function of }x
\\\text{Domain: }
\left[ -\dfrac{7}{4},\infty \right)
\\\text{NOT a linear function}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
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}