### Instruction: Find the domain of the function
\[h(x) = \frac{3x - 1}{\sqrt{x - 5}}.\].### Response: To find the domain of a function, I need to identify all the values of x that make the function well-defined, that is, that don't cause any division by zero or square roots of negative numbers.
In this case, the denominator of the fraction has a square root, so I want to avoid any values of x that make the expression inside the square root negative.
To do that, I can set up an inequality and solve for x:
\[\begin{aligned}
x - 5 &\geq 0 \\
x &\geq 5
\end{aligned}\]
This means that any value of x that is greater than or equal to 5 is allowed in the domain of h.
On the other hand, the numerator of the fraction doesn't have any restrictions, since any value of x will make 3x - 1 a valid number.
Therefore, the domain of h is the set of all real numbers that are greater than or equal to 5.
I can write this using interval notation as [5, infinity), or using set-builder notation as {x | x >= 5}.