Let's analyze the function $f(x) = \frac{1}{\sqrt{x - 3}}$ to determine its domain and range.

Domain:

The domain of a function consists of all possible input values (x-values) for which the function is defined.

1. Square Root Condition: The square root function, $\sqrt{x - 3}$, is only defined when $x - 3 \geq 0$. Therefore, $x \geq 3$.

2. Denominator Condition: The function also has a denominator, so $\sqrt{x - 3}$ cannot be zero. This occurs when $x - 3 \neq 0$, i.e., $x \neq 3$.

Combining these two conditions, we get $x > 3$. Therefore, the domain of $f(x)$ is:

$\text{Domain} = (3, \infty)$

Range:

The range of a function is the set of all possible output values (y-values) that the function can take.