The problem is to determine the domain of the function $f(x) = \frac{1}{2\sqrt{x-3}}$.

1. Problem Description

The problem is to determine the domain of the function $f(x) = \frac{1}{2\sqrt{x-3}}$.

2. Solution Steps

To find the domain of the function $f(x) = \frac{1}{2\sqrt{x-3}}$, we need to consider two conditions:

(1) The expression inside the square root must be non-negative.

(2) The denominator must not be equal to zero.

For the first condition, we must have $x - 3 \ge 0$, which means $x \ge 3$.

For the second condition, since the square root is in the denominator, we must have $2\sqrt{x-3} \ne 0$, which implies $\sqrt{x-3} \ne 0$, and thus $x-3 \ne 0$, so $x \ne 3$.

Combining both conditions, we require $x \ge 3$ and $x \ne 3$. This means $x > 3$. Therefore, the domain of the function is all real numbers greater than

3.

3. Final Answer

$x > 3$