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The square root function appears in many mathematical expressions and plays a key role in determining the domain of functions. A square root function includes an expression that contains the symbol \( \sqrt{} \). It's vital because a square root is only defined when the value inside, known as the radicand, is non-negative. For example, in the function \( g(x) = 9 - \sqrt{x + 103} \), the radicand is \( x + 103 \).To find the domain of a function involving a square root, we must ensure the radicand is greater than or equal to zero. So, for any function \( \sqrt{x} \), the domain consists of all \( x \) values that make the expression inside the root non-negative. This is often the first step in analyzing functions with square roots, ensuring it only admits real number inputs.

Inequality solving is an essential skill, particularly when finding the domain of functions involving radicals or other expressions with restrictions.To solve inequalities like the one in the exercise \( x + 103 \geq 0 \), we'll first isolate the variable \( x \). Subtract 103 from both sides to maintain balance, which gives:\[x \geq -103\]Solving inequalities for functions gives us the set of inputs (or domain) for which the function is defined. By treating the inequality similarly to an equation, but respecting its direction (whether \( \geq \), \( \leq \), \( > \), or \( < \)), we make sure we capture all valid inputs for the function.

Interval notation provides a concise way to represent subsets of the real number line, especially useful after solving inequalities.From the exercise's solved inequality \( x \geq -103 \), we can express this domain in interval notation as \( [-103, \infty) \). Here's how it breaks down:

• \(-103\) is where the domain starts, and it includes this number, hence the square bracket \([\).
• The infinity symbol \( \infty \) implies that the domain extends indefinitely to the right, but it does not include \( \infty \) itself, hence the round parenthesis \()\).

Interval notation efficiently communicates the range of possible values for \( x \), simplifying how we understand and transmit solutions to inequalities.