To determine the domain of the function [tex]\( f(x) = \sqrt{\frac{1}{2} x - 10} + 3 \)[/tex], we need to ensure that the expression inside the square root is non-negative because the square root of a negative number is not defined within the real numbers.
Given the function [tex]\( f(x) = \sqrt{\frac{1}{2} x - 10} + 3 \)[/tex], the expression inside the square root, [tex]\( \frac{1}{2} x - 10 \)[/tex], must be greater than or equal to zero. This forms the basis of the inequality we need to solve:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
Next, we solve this inequality step-by-step:
1. Isolate [tex]\( x \)[/tex] on one side of the inequality:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
[tex]\[ \frac{1}{2} x \geq 10 \][/tex]
2. Multiply both sides of the inequality by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[ x \geq 20 \][/tex]
Therefore, the inequality that can be used to find the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
This ensures that the expression inside the square root is non-negative, thereby making the function [tex]\( f(x) \)[/tex] defined. So, the correct answer is:
[tex]\[ \frac{1}{2} x - 10 \geq 0 \][/tex]
