91Ó°ÊÓ

The square root function is a function of the form \( f(x) = \sqrt{g(x)} \). This means that we take the square root of an expression. The most important thing to remember about square root functions is that the expression inside the square root must be non-negative. In other words, the value under the square root must be zero or positive. This is because we cannot take the square root of a negative number in real numbers. For example, if we have \( f(x) = \sqrt{9 - x^2} \), then the expression under the square root, \( 9 - x^2 \), has to be zero or greater.

Solving inequalities is a fundamental skill in algebra. When you solve an inequality, you are looking for the set of values that satisfy it. In the context of our exercise, we need to ensure the expression within the square root is non-negative. Let's walk through the solution step by step:

• First, we establish the inequality: \( 9 - x^2 \geq 0 \)
• Second, rearrange the inequality to isolate \( x \): \( x^2 \leq 9 \)
• Third, solve for \( x \) by taking the square root on both sides, considering both positive and negative solutions: \( -3 \leq x \leq 3 \)

By solving the inequality, we determine the range of values for \( x \) that keep the expression under the square root non-negative.

The domain of a function is the set of all possible input values (or \( x \) values) that the function can accept. When dealing with square root functions, identifying the domain involves ensuring that the expression within the square root is non-negative. In simpler terms, we need to find the values of \( x \) that make the expression inside the square root non-negative.
For our function \( f(x) = \sqrt{9 - x^2} \), we determined earlier that solving the inequality \( 9 - x^2 \geq 0 \) gives us the values \( -3 \leq x \leq 3 \).
Therefore, the domain of our function is all real numbers from -3 to 3, inclusive. This means that any value of \( x \) within this interval will make our function valid.