Let's find the domain and range of the function $f(x) = \frac{1}{\sqrt{x - 3}}$.

Domain:

The domain of a function consists of all the values of $x$ for which the function is defined. In this case, the function involves a square root and a division.

1. Square Root Condition: The expression inside the square root, $x - 3$, must be non-negative, because the square root of a negative number is not defined in the set of real numbers. Therefore, $x - 3 \geq 0$, which simplifies to: $x \geq 3$

2. Division Condition: The expression under the square root, $\sqrt{x - 3}$, must not be zero because division by zero is undefined. Therefore: $\sqrt{x - 3} \neq 0$ This implies $x - 3 \neq 0$, so $x \neq 3$.

Combining these two conditions, $x$ must be greater than 3: [ \text{Domain: } x > 3 \