Step 1: Identify the conditions for the function to be defined in real numbers. The given function is $f(x) = \sqrt{\frac{2}{x-2}} - 3$. For $f(x)$ to be a real number, two conditions must be met: 1. The expression under the square root must be non-negative: $\frac{2}{x-2} \ge 0$. 2. The denominator of the fraction cannot be zero: $x-2 \ne 0$. Step 2: Apply the condition for the denominator. From $x-2 \ne 0$, we know that $x \ne 2$. Step 3: Apply the condition for the square root. For $\