To determine the domain of the function \( f(x) = \sqrt{\frac{1}{2} x - 10} + 3 \), we need to ensure that all values under the square root are non-negative because the square root of a negative number is not defined in the set of real numbers.
Given the function \( f(x) = \sqrt{\frac{1}{2} x - 10} + 3 \), the key part we need to focus on is \( \sqrt{\frac{1}{2} x - 10} \). For the square root to be defined, the expression inside the square root, \( \frac{1}{2} x - 10 \), must be greater than or equal to zero.
Therefore, we set up the following inequality:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
This inequality ensures that the value inside the square root is non-negative. Solving this inequality will give us the domain of the function.
Thus, the correct inequality that can be used to find the domain of \( f(x) \) is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
So the correct answer is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
