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When you determine the domain of a function, using interval notation can make it easy to understand. Interval notation is a mathematical way of writing down a set of numbers along an interval.
It describes where a variable like \( x \) can exist on a number line.

• The symbol \([a, b]\) represents all the numbers from \( a \) to \( b \) including \( a \) and \( b \).
• Parentheses \((a, b)\) are used to show that \( a \) and \( b \) are not included in the interval.

For infinite intervals, you use \( \infty \) to indicate that the interval extends indefinitely. For example, \([2, \infty)\) means \( x \) is greater than or equal to 2 and can go on forever.
Once you solve for the inequality, it is crucial to translate the solution into this notation to provide a clear understanding of the domain of the function.

Square root functions are special functions that involve a square root symbol. The general form looks something like \( f(x) = \sqrt{x} \). In the context of the example function \( f(x) = 3 \sqrt{x - 2} \), there are specific rules concerning the input values.
A square root operation requires that whatever is inside the square root be non-negative. Meaning, the expression inside the square root should always be zero or more because you can't take the square root of a negative number in the set of real numbers.

• For \( \sqrt{x - 2} \), \( x - 2 \) must be greater than or equal to zero.

This requirement is fundamental and is a part of determining the domain. Anytime you have a square root function like this, start with the expression under the root to find acceptable \( x \) values.

Solving inequalities is a bit like solving equations, but with a few more considerations. The inequality \( x - 2 \geq 0 \) expresses a range of values for \( x \) that satisfies the condition of the function.
Unlike an equation, where you find specific value(s), inequalities describe a range. Here’s a quick way to solve this particular inequality:

• Add 2 to both sides of \( x - 2 \geq 0 \).
• You’ll get \( x \geq 2 \).

This solution tells you that any \( x \) value starting from 2 and going upwards fits the inequality. This kind of solution gives you insight into not single points but entire intervals of acceptable inputs that keep the function's output real.

Function restrictions are vital because they tell us which values a function can and cannot take. In square root functions, like \( f(x) = 3 \sqrt{x - 2} \), these restrictions are usually based on the need for the expression inside the root to remain non-negative.
If you were to go against these restrictions and try to use a value that doesn't satisfy the condition (such as putting \( x = 1 \)), the function would become undefined in the real number system. This can lead to errors or misunderstandings.

• Restrictions arise naturally from the function’s mathematical form.
• They serve as a guide to what values make sense for the function in the context of real numbers.

Recognizing function restrictions is essential because it allows you to accurately determine the domain and safely know how the function behaves.