To find the domain of the function $g(x) = \log_3(-x - 2)$, we need to consider the condition for the logarithmic function to be defined.

The logarithm of a number is only defined when the argument is positive. In this case, the argument is $-x - 2$, so we need:

$-x - 2 > 0$

Now, solve for $x$:

$-x > 2$

Multiplying both sides by $-1$ (and remembering to flip the inequality sign):

$x < -2$

Thus, the domain of $g(x)$ is $x < -2$, or in interval notation:

$(-\infty, -2)$

Summary:

The domain of the function $g(x) = \log_3(-x - 2)$ is $(-\infty, -2)$.

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Would you like more details on logarithmic functions, or do you have any other questions? Here are some related questions:

1. How do you graph the function $g(x) = \log_3(-x - 2)$?
2. What is the range of the function $g(x) = \log_3(-x - 2)$?
3. How does the base of the logarithm affect its graph?
4. How do you solve logarithmic equations like $\log_3(x + 2) = 4$?
5. Can you find the domain of a logarithmic function with multiple terms inside the logarithm?

Tip: When working with logarithmic functions, always check the domain to ensure the argument is positive, as logarithms are undefined for non-positive values.