Problem 27 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{\sqrt{x-2}}{x-5}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{\sqrt{x-2}}{x-5}\) is \([2,5) \cup (5, +\infty)\)

Step 1: Analyze the Square Root

When dealing with square roots, it's important to understand that the term under the square root (the radicand) must be non-negative (i.e., greater than or equal to zero) because the square root of a negative number is not a real number. So first, we must find the x values for which \(x-2 \geq 0\). Solving this inequality, we find \(x \geq 2\)

Step 2: Analyze the Denominator

For a function to be defined, the denominator must not be zero, as division by zero is undefined. So, we must find the x values for which \(x-5 \neq 0\). Solving this, we find that \(x \neq 5\)

Step 3: Combine the Conditions

Since both conditions (from steps 1 and 2) must be met, the domain of the function is the set of all x values that satisfy both conditions. Therefore, the domain is all real numbers greater than or equal to 2, except 5. This can be written in interval notation as \([2,5) \cup (5, +\infty)\)