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A square root function involves finding the square root of a given expression. For instance, take the function given in the original exercise: \( g(x) = \sqrt{2 - 3x} \). Here, the expression inside the square root is \(2 - 3x\).

A key aspect of square root functions is that the expression inside the square root must be non-negative. This is because the square root of a negative number is not a real number (it's complex).

To determine the domain of a square root function, you need to find where the expression inside the square root is greater than or equal to zero. This leads to setting up an inequality. Let's move on to that concept next.

Solving inequalities is a crucial step in finding the domain of many functions, including square root functions. In our example, the inequality to solve is \(2 - 3x \geq 0\).

Let's break down the solution step by step:

• First, identify the inequality: \(2 - 3x \geq 0\).
• Next, isolate the variable by subtracting or adding terms. Here, subtract \(2\) from both sides: \(-3x \geq -2\).
• Finally, divide by \(-3\) to solve for \(x\). Remember to reverse the inequality sign when dividing by a negative number: \(x \leq \frac{2}{3}\).

This tells us that \(x\) must be less than or equal to \(\frac{2}{3}\) for the function \(g(x)\) to be real and defined. Understanding inequality solving is essential for many mathematical problems beyond finding domains.

Interval notation is a way to describe a range of values for a variable. It uses parentheses \(()\) and brackets \([]\) to show if endpoints are included or not. For example, \((a, b)\) represents values strictly between \(a\) and \(b\), not including \(a\) or \(b\).

If endpoints are included, use brackets: \([a, b]\). Combining them, such as \((a, b]\), includes \(b\) but not \(a\).

From our inequality solution \(x \leq \frac{2}{3}\), the domain is all values less than or equal to \(\frac{2}{3}\).

In interval notation, this is written as \((-\infty, \frac{2}{3}]\). Here, \(-\infty\) means it goes infinitely in the negative direction and \(\frac{2}{3}\) is included in the set of values for \(x\).

Mastering interval notation helps to clearly communicate solution sets, especially in higher mathematics.