To find the domain of the function $g(x) = \frac{x + 6}{x - 2}$, we need to determine the values of $x$ that make the function undefined.

Step-by-Step Solution:

1. Identify the Denominator: The function is undefined when the denominator equals zero, since division by zero is undefined.

2. Set the Denominator to Zero: $x - 2 = 0$ Solving this equation gives: $x = 2$

3. Domain in Interval Notation: Since the function is undefined at $x = 2$, we exclude $2$ from the domain. Therefore, the domain of $g(x)$ is all real numbers except $x = 2$. In interval notation, this is written as: $(-\infty, 2) \cup (2, \infty)$

Answer:

The correct choice is: $x \in (-\infty, 2) \cup (2, \infty)$

Would you like further clarification on any part of this?

Here are some related questions to explore:

1. How would the domain change if the function were $g(x) = \frac{x + 6}{x - 3}$?
2. What is the range of $g(x) = \frac{x + 6}{x - 2}$?
3. How can you find the vertical asymptote of a rational function like $g(x) = \frac{x + 6}{x - 2}$?
4. How would the domain of $g(x) = \frac{x + 6}{x - 2}$ be expressed using set notation instead of interval notation?
5. Can $g(x)$ have a horizontal asymptote, and if so, how do you find it?

Tip: For rational functions, always set the denominator to zero to find values excluded from the domain.