Final answer:
The domain of the function g(x) = log₃(x² - 4) is the set of x-values where the argument of the logarithm is greater than zero. After solving the inequality x² - 4 > 0, it is determined that the domain is x in (-∞, -2) ∪ (2, ∞), which corresponds to option a).
Step-by-step explanation:
The domain of a function is the set of all possible inputs (x-values) for which the function is defined. For the function g(x) = log₃(x² - 4), the argument of the logarithm, x² - 4, must be greater than zero because the logarithm of a negative number or zero is undefined. Therefore, we must solve the inequality x² - 4 > 0.
To find the domain, factor the quadratic equation: (x + 2)(x - 2) > 0. This gives us two critical points, x = -2 and x = 2, where the expression changes sign. We can then set up a sign chart or test values in the intervals (-∞, -2), (-2, 2), and (2, ∞) to find where the expression is positive.
Upon examination, the expression x² - 4 is positive when x is less than -2 or greater than 2. Thus, the domain of g(x) is x in (-∞, -2) ∪ (2, ∞), corresponding with option a).