Problem 28 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{\sqrt{x-3}}{x-6}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{\sqrt{x-3}}{x-6}\) is \(x \geq 3\) and \(x \neq 6\).

Step 1: Solve the inequality under the square root

First, we must solve the inequality \(x-3\geq0\) as this forms the first part of our domain. When we add 3 to both sides we get \(x\geq3\). This means that x can be any number greater than or equal to 3 for the square root to be a real number.

Step 2: Find the value that makes the denominator equal to zero

Next, we set the denominator equal to zero and solve for x to find the value that we must exclude from our domain: \(x-6=0\). Solving this equation gives us \(x=6\). Therefore, x cannot be equal to 6 because this would make our function undefined.

Step 3: Put together the complete domain

Taking into account these considerations, the domain of the function \(g(x) = \frac{\sqrt{x-3}}{x-6}\) is \(x \geq 3\) and \(x \neq 6\). That means x can be any real number greater than or equal to 3 but not equal to 6.