Problem 62 - Beginning and Intermediate Algebra

Question

How do you find the domain of a square root function?

Step-by-Step Solution

Answer

To find the domain of a square root function \(f(x) = \sqrt{g(x)}\), set the expression inside the square root as non-negative (\(g(x) \geq 0\)) and solve the inequality. The solution will give the range of x-values for which the function is defined. Write the domain in interval notation.

Step 1: 1. Identify the square root function

First, take a look at the function and identify it as a square root function. A square root function typically has the form: \(f(x) = \sqrt{g(x)}\)

Step 2: 2. Set the expression inside the square root as non-negative

Next, we want to ensure that the expression inside the square root, \(g(x)\), is non-negative. This is because the square root function is only defined for non-negative inputs. So we set up the inequality: \(g(x) \geq 0\)

Step 3: 3. Solve the inequality

Now, solve the inequality (\(g(x) \geq 0\)). The solution to this inequality will give the range of x-values for which the function is defined. Remember that if the inequality is a quadratic or higher-order polynomial, always check for critical points and intervals.

Step 4: 4. Write the domain

Finally, write down the domain of the function using the solution of the inequality. The domain is usually expressed in interval notation, such as \((a, b)\), \([a, b)\), or \((-\infty, a] \cup [b, \infty)\), depending on the nature of the solution. Note: If the given function is a more complex expression that involves multiple square roots, be sure to consider each square root expression separately while solving the inequality. The overall domain of the function will be the intersection of the individual domains.