Problem 19 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{1}{\sqrt{x-3}}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)\) is (3, +\(\infty\)).

Step 1: Identify the restrictions

First, recognize that \(x-3\) must be greater than 0 because it is under the square root and we cannot take a square root of a negative number. Secondly, since it is in the denominator, it cannot be equal to 0 because division by zero is undefined. Therefore, the inequality to solve is \(x-3 > 0\).

Step 2: Solve the inequality

Solve the inequality by adding 3 to both sides: \(x > 3\).

Step 3: Write the domain

The domain of the function \(g(x)\) is all real numbers greater than 3. In interval notation, this is represented as (3, +\(\infty\)).