Problem 28 - Precalculus 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 | x \(\geq\) 3 and x \(\neq\) 6}

Step 1: Set the Value Under the Square Root Greater than or Equal to Zero

Find the x-values that make x - 3 \(\geq\) 0. Solving this inequality leads to x \(\geq\) 3.

Step 2: Determine the x-values that Make the Denominator Not Equal to Zero

Set the denominator of the function not equal to zero, then solve for x to find the x-values. This gives x \(\neq\) 6.

Step 3: Find the Intersection of the Values from Step 1 and Step 2

The domain is the intersection of all x-values that satisfy both condition from step 1 and step 2. This gives us {x | x \(\geq\) 3 and x \(\neq\) 6}. This set notation expresses the domain of the function. The vertical bar | stands for 'such that'.