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A square root function involves taking the square root of a number or an expression. The core rule to remember is that the number or expression inside the square root, called the radicand, must be greater than or equal to zero. This is because the square root of a negative number is not defined within the set of real numbers. For instance, in the function \( \sqrt{x} \), the radicand is \( x \), and it must satisfy \( x \geq 0 \).
Similarly, if you have a more complex expression like \( \sqrt{1-x} \), the condition slightly changes. The radicand \( 1-x \) should be non-negative, setting up the inequality \( 1-x \geq 0 \), which simplifies to \( x \leq 1 \).
In any mathematical function involving square roots, it is crucial to establish these conditions to ensure we only consider valid, real-number results.

Inequalities are mathematical statements that compare two values or expressions. They are established using symbols like \( \geq \) (greater than or equal to) and \( \leq \) (less than or equal to).
When dealing with inequalities, we seek to identify all possible values that satisfy the given conditions. In the context of our square root function, two inequalities must be satisfied: \( x \geq 0 \) and \( x \leq 1 \). Contemplating both conditions together gives us a range of values that are possible as inputs, namely, values of \( x \) that fulfill both criteria at the same time.
By solving inequalities, we figure out the domain of functions and often visualize them on a number line to understand which sections are included based on the given conditions.

Interval notation is a shorthand method used to describe a set of numbers within a certain range. It is especially helpful when expressing the domain of a function as it clearly shows the start and end points.
For example, if we found through solving inequalities that a function's possible input values (domain) range from 0 to 1, we write this as \([0, 1]\). The use of square brackets \([\ ])\) indicates that both endpoints are included in the set, known as a closed interval.
Here’s a quick guide to interval notation:

• \((a, b)\) refers to an interval excluding \(a\) and \(b\).
• \([a, b)\) includes \(a\) but not \(b\).
• \((a, b]\) includes \(b\) but not \(a\).
• \([a, b]\) includes both \(a\) and \(b\).

Understanding interval notation is vital because it succinctly communicates which numbers belong to the domain or range of a function.