Problem 27 - Precalculus 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 \([2, 5)\) and \((5, \infty)\)

Step 1: Condition for the square root

The square root function \(\sqrt{x - 2}\) is defined for all values, where the radicand \(x - 2\) is greater than or equal to zero. So, let's solve for x. Set \(x - 2 \geq 0\), Adding 2 to both sides gives \(x \geq 2\)

Step 2: Condition for the denominator

The function is undefined if the denominator is zero. The denominator in this case is \(x - 5\), which should be different from zero. Solving \(x - 5 \neq 0\) gives \(x \neq 5\)

Step 3: Intersection of both conditions

Now, taking both these conditions together, the function is defined for \(x \geq 2\), but \(x \neq 5\). So, the function is defined for all \(x\) from 2 to 5 and x greater than 5