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Interval notation is a way of representing the set of all possible values that a variable can take. It uses brackets and parentheses to show which numbers are included or excluded from the set.

For example, when we say \((-\infty, -2) \cup (-2, 1]\), we're describing two intervals. The first interval \((-\infty, -2)\) represents all numbers less than -2. Since "\((-\infty, -2)\)" uses an open parenthesis, it excludes -2 itself.

The second interval \((-2, 1]\) includes all numbers greater than -2 up to and including 1. The square bracket "\([\)" indicates that 1 is included in the interval. The union symbol "\(\cup\)" connects these two intervals, meaning that the domain consists of all numbers in either of those sets.

Inequalities are mathematical expressions that show the relationship between two values, telling us if one value is larger, smaller, or not equal to another.

In the function \(rac{\sqrt{1-x}}{x^{2}-4}\), we needed to consider the inequality \((1 - x) \geq 0\) to ensure the term inside the square root is not negative. Solving this inequality gives us \(x \leq 1\).

This tells us that x can take any value less than or equal to 1. Solving inequalities is key in determining the domain of functions, as it helps identify the range of permissible values for the variables involved.

Square root functions involve a root expression, such as \(\sqrt{1-x}\), and require special consideration when determining the domain.

The expression inside the square root (1-x) must be zero or positive, as the square root of a negative number is not defined in the real number system.

We find this by setting up the inequality \((1-x) \geq 0\), leading us to \(x \leq 1\). This inequality tells us that all values must be less than or equal to 1 to ensure the function is valid.

Understanding square root functions is crucial in determining domains and ensuring the expressions are mathematically valid.

Rational functions are functions that involve fractions of polynomials, like \(\frac{\sqrt{1-x}}{x^{2}-4}\).

The denominator \(x^2 - 4\) should never be zero, as it would make the function undefined. To find where the denominator equals zero, we solve \(x^2-4=0\) by factoring, obtaining \(x = 2\) or \(x = -2\).

These values must be excluded from the domain. By understanding rational functions, we learn to carefully handle the values that make the denominator zero to maintain the function's validity.

These steps ensure that the function remains defined across its domain.