To determine the domain of the function [tex]\( f(x) = \sqrt{\frac{1}{2} x - 10} + 3 \)[/tex], we need to ensure that the quantity inside the square root is non-negative. In mathematical terms, the expression inside the square root must be greater than or equal to zero, because the square root of a negative number is not defined in the set of real numbers.
We start by focusing on the expression inside the square root:
[tex]\[ \sqrt{\frac{1}{2} x - 10} \][/tex]
For this expression to be defined, we require that:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
Next, we solve this inequality step by step:
1. Isolate [tex]\( x \)[/tex]:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
2. Add 10 to both sides:
[tex]\[ \frac{1}{2} x \geq 10 \][/tex]
3. Multiply both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[ x \geq 20 \][/tex]
So, the inequality that represents the domain of the function [tex]\( f(x) = \sqrt{\frac{1}{2} x - 10} + 3 \)[/tex] is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
Thus, the correct choice from the given options is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
