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The domain of a function is a fundamental concept in mathematics. It represents the complete set of values for which a function is defined. When working with functions, it's essential to identify their domain to understand where they "live" on the number line. For example, in a function like \( f(x) = \sqrt{x^2 - 4} \), the expression under the square root affects the domain. Since square roots of negative numbers aren't real, we need \( x^2 - 4 \geq 0 \).

To find the domain:

• First, set the expression inside the square root to be non-negative: \( x^2 - 4 \geq 0 \).
• Solve for \( x \) to determine the values that meet this condition.

This helps us find the valid range of \( x \) that will produce real output from the function. Once we find these values, we can describe the domain in interval notation. For our example, the domain is \((-\infty, -2] \cup [2, +\infty)\), clearly showing where the function is defined.

Inequalities express the relative size or order of two values and are crucial in determining domains. With an inequality like \( x^2 - 4 \geq 0 \), we are looking to find ranges of \( x \) that satisfy the inequality.

Steps to solve:

• Start by factoring the expression: \( (x-2)(x+2) \geq 0 \).
• Identify critical points where each factor could cause the product to change sign. For \( (x-2)(x+2) \), these points are \( x=2 \) and \( x=-2 \).
• Use these points to divide the number line into intervals.
• Test each interval to determine where the inequality holds true.

By testing intervals, we can ensure each is correctly classified as either satisfying or not satisfying the inequality. This approach allows us to find the intervals for \( x \) in \( f(x) = \sqrt{x^2 - 4} \) where the function is defined and continuous.

The square root function, represented as \( \sqrt{x} \), is a common mathematical function that involves finding a number which, when multiplied by itself, gives \( x \). It's defined for non-negative numbers, which is why its domain is crucial in practical scenarios.

Key aspects include:

• It requires the input (under the root) to be zero or positive for the square root to yield a real number.
• The square root function is continuous in its domain, meaning it doesn't "break" at any point within its valid range.

In our function \( f(x) = \sqrt{x^2 - 4} \), this translates into needing \( x^2 - 4 \geq 0 \). By solving this inequality, we determine the intervals \( (-\infty, -2] \) and \( [2, \infty) \) where the function is both defined and continuous. Understanding these concepts allows you to grasp why and how the square root function applies in practice.