Answer
The given relation defines $y$ as a function of $x$.
domain: $(-\infty, 0) \cup (0, +\infty)$
range: $(-\infty, 0) \cup (0, +\infty)$
Work Step by Step
Solve for $y$ to obtain:
$xy = -6
\\\frac{xy}{x} =\frac{-6}{x}
\\y=-\frac{6}{x}$
This means that the given equation is equivalent to $y=-\frac{6}{x}$.
The equation above will give only one value of $y$ for every value of $x$. This means that each $x$ is paired with only one value of $y$.
Thus, the given relation defines $y$ as a function of $x$.
Note that in $y=-\frac{6}{x}$, the value of $x$ cannot be zero because it will make the expression undefined. This means that the domain of the given function is the set of real numbers except $0$. In interval notation, the domain is $(-\infty, 0) \cup (0, +\infty)$.
Note that when $-6$ is divided by any non-zero number, the quotient will never be zero. Thus, the value of $y$ can be any real number except zero. In interval notation, the range is $(-\infty, 0) \cup (0, +\infty)$.
