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Understanding quadratic inequalities is vital when dealing with functions like \( g(x) = \sqrt{x^2 - 2x - 8} \). Here, the expression inside the square root must be non-negative; hence, we solve the inequality \( x^2 - 2x - 8 \geq 0 \).

A quadratic inequality involves expressions of the form \( ax^2 + bx + c \leq 0 \), \( ax^2 + bx + c \geq 0 \), and their strict forms. The solutions to these inequalities are generally found using the roots of the associated quadratic equation, which in our case are \(x = 4\) and \(x = -2\). These roots are found using techniques like factoring or the quadratic formula.

Once you have the roots, the number line can be divided into intervals. Each interval is tested to determine if it satisfies the inequality. For \( g(x) = \sqrt{x^2 - 2x - 8} \), the inequality holds true for \( x \leq -2 \) and \( x \geq 4 \), forming the basis of the domain for the function. Understanding how intervals work in relation to inequalities is crucial for finding function domains.

Square root functions are defined only for non-negative values. This limitation requires us to ensure that the expression under the square root, or "radicand," is zero or positive.

For the function \( g(x) = \sqrt{x^2 - 2x - 8} \), the radicand \( x^2 - 2x - 8 \) needs to be evaluated such that it meets this criterion. If \( x^2 - 2x - 8 \) turns out to be negative, \( g(x) \) cannot produce a real number. Hence, identifying the proper domain involves solving the inequality \( x^2 - 2x - 8 \geq 0 \).

When working with square root functions, always pay attention to the values that can turn the radicand negative. This is key in correctly identifying the domain. For \( g(x) \), the domain is \(( -\infty, -2 ] \cup [ 4, \infty )\), where these intervals guarantee that the radicand is non-negative, allowing the square root to produce real and valid outputs.

Factoring quadratics is a powerful technique when solving equations and inequalities. It helps in simplifying quadratic expressions into their linear components, making them easier to analyze.

For the quadratic expression \( x^2 - 2x - 8 = 0 \), we seek factors that multiply to \(-8\) and add up to \(-2\). These factors are \( (x-4) \) and \( (x+2) \), since \(-4\) and \(+2\) multiply to enter \(-8\) and sum to \(-2\).

By setting these factors equal to zero, we find the roots \(x = 4\) and \(x = -2\). These roots are critical for establishing intervals on the number line where the quadratic inequality holds. Factoring is often faster and more straightforward than using the quadratic formula for solving simple quadratics. However, it's important to always check your work. Incorrect factors can lead to wrong interval conclusions and, subsequently, an incorrect domain for functions like \( g(x) = \sqrt{x^2 - 2x - 8} \).