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Inequalities are mathematical statements that compare two values, showing that one is larger or smaller than the other. In the context of function domains, we often use inequalities to determine the set of input values (usually represented by \(x\)) for which the function is defined. For instance, when dealing with square root functions, we typically set up inequalities to ensure the radicand (the expression inside the square root) is non-negative, because the square root of a negative number is not defined in the realm of real numbers.

To solve an inequality, you need to find all values that satisfy it. This might involve various algebraic manipulations, including factoring or using formulas like the quadratic formula for more complex expressions. Once solved, inequalities provide insights into the intervals over which a function is valid, offering a crucial step in understanding the function's behavior.

Quadratic expressions are polynomials of degree two, typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These expressions often describe a parabolic shape on a graph. For example, the expression \(2x^2 + 3\) represents a parabola that opens upwards, given \(a = 2\), which is positive.

Analyzing the behavior of a quadratic expression is vital when working with functions like \(g(x) = \sqrt{2x^2 + 3}\). Since the parabola "lives" above the x-axis for all \(x\), this tells us that \(2x^2 + 3\) is always non-negative—that is, there are no real roots where the expression equals zero or becomes negative. This insight is helpful, especially when determining the domain of functions involving square roots. Recognizing the general behavior of quadratic expressions simplifies finding intervals that make the corresponding function valid.

Interval notation is a mathematical notation used to describe a range of numbers. It's concise and clear, helping to indicate which numbers are included within an interval on the number line. There are different types of interval notation, including open intervals, denoted by \((a, b)\), which exclude the endpoints, and closed intervals, denoted by \([a, b]\), which include the endpoints. If the interval extends indefinitely in one direction, you might see \((-\infty, a)\) for infinity or \((b, +\infty)\).

For the function \(g(x) = \sqrt{2x^2 + 3}\), the inequality analysis revealed that the function is defined for all real numbers. Therefore, the domain in interval notation is \(( -\infty, +\infty )\), indicating the function has no restrictions on the values \(x\) can take. Interval notation simplifies the presentation of complex sets of numbers, making them easy to read and interpret in mathematical contexts.