Intermediate Algebra, Chapter 3, 3.5, Section 3.5, Problem 54

Identify whether the relation $\displaystyle y = \frac{7}{x-2}$ defines $y$ as the function of $x$ and give the domain.

Given any value of $x$ in the domain, we find $y$ by substituting $2$ from $x$ and then dividing the result into $7$.
This process produces exactly one value of $y$ for each value in the domain, so the given equation defines a function.
The domain includes all real numbers except those which value the denominator . We find these numbers by setting the
denominator equal to and solving for $x$. In this case,

$
\begin{equation}
\begin{aligned}
x - 2 &= 0 && \text{Add 2}\\
\\
x &= 2
\end{aligned}
\end{equation}
$

The domain includes all real numbers except $2$, written $(-\infty, 2) \bigcup (2, \infty)$