Answer
The domain of the function $g\left( x \right)=\frac{\sqrt{x-2}}{x-5}$ is $\left[ 2,5 \right)\cup \left( 5,\infty \right)$.
Work Step by Step
Now, consider the function $g\left( x \right)=\frac{\sqrt{x-2}}{x-5}$.
We can see that this function contains division and division by 0 is not defined. Exclude those values of $x$ from the domain that cause the denominator to be zero.
Thus, set the denominator equal to 0, that is, $x-5=0$.
This implies that $x=5$.
So, exclude 5 from the domain.
Next, note that this function also contains the square root in the numerator and the square root of a negative function is not defined; therefore, the term written inside the square root must be non-negative. That is, $x-2\ge 0$ or $x\ge 2$.
Therefore, the domain of the function $g\left( x \right)=\frac{\sqrt{x-2}}{x-5}$ is $\left[ 2,5 \right)\cup \left( 5,\infty \right)$.
