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Logarithmic functions are mathematical functions that help us solve problems involving exponents. The function \( g(x) = \log_{3}(x^2 - 1) \) is an example of a logarithmic function. Here, \( \log_{3} \) indicates a logarithm with a base of 3. The term inside, or the **argument**, of the logarithm is \( x^2 - 1 \).

For any logarithmic function, its argument must be positive. This means \( x^2 - 1 \) should be greater than 0 for the function to be valid. If the inside of the logarithm isn't positive, the function becomes undefined. This is key when considering the domain of a function, as we'll only include x-values in our domain where the argument stays positive.

Inequalities are mathematical expressions involving symbols like greater than \( (>) \), less than \( (<) \), or equals \( (=) \). Solving inequalities helps us find out which values of \( x \) are permissible.

In our function \( g(x) = \log_{3}(x^2 - 1) \), the inequality we need to focus on is \( x^2 - 1 > 0 \).

• First, equate \( x^2 - 1 \) to zero: \( x^2 - 1 = 0 \).
• Solve this equation to find the critical points \( x = \pm 1 \).

These critical points are where the expression shifts from positive to negative or vice versa.

Evaluating the inequality, we find \( x^2 - 1 > 0 \) holds for \( x < -1 \) and \( x > 1 \). This solution helps us define the domain of the function later.

Interval notation is a method of writing sets of numbers, particularly to describe the domain of functions. In the context of our logarithmic function \( g(x) = \log_{3}(x^2 - 1) \), interval notation helps us express all possible values of \( x \) based on the conditions we've derived from the inequality.

After solving \( x^2 - 1 > 0 \), we find that values of \( x \) satisfying this inequality are in the two intervals \( (\infty, -1) \) and \( (1, \infty) \).

• \( (-\infty, -1) \) covers all values less than \(-1\).
• \( (1, \infty) \) covers all values greater than \(1\).

The use of round brackets indicates that the endpoints \(-1\) and \(1\) are not included, as the inequality is strictly "greater than" and not "greater than or equal to." Therefore, the domain in interval notation is denoted as \( (-\infty, -1) \cup (1, \infty) \), clearly showing which values make the logarithmic function valid.