To determine the domain of the function [tex]\( f(x) = \sqrt{x-3} \)[/tex], we need to ensure that the expression inside the square root is non-negative, because the square root function is only defined for non-negative numbers in the real number system.
Let's analyze the function step-by-step:
1. The function [tex]\( f(x) = \sqrt{x-3} \)[/tex] involves a square root.
2. For the square root function to be defined, the argument inside the square root must be greater than or equal to zero. This means [tex]\( x - 3 \)[/tex] must be non-negative.
To find the values of [tex]\( x \)[/tex] that satisfy this condition, we solve the following inequality:
[tex]\[ x - 3 \geq 0 \][/tex]
To solve for [tex]\( x \)[/tex], we simply add 3 to both sides of the inequality:
[tex]\[ x \geq 3 \][/tex]
This tells us that the domain of the function [tex]\( f(x) = \sqrt{x-3} \)[/tex] consists of all [tex]\( x \)[/tex] values that are greater than or equal to 3.
Therefore, the correct inequality that can be used to find the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ x - 3 \geq 0 \][/tex]
Among the given options:
1. [tex]\(\sqrt{x-3} \geq 0\)[/tex]
2. [tex]\(x-3 \geq 0\)[/tex]
3. [tex]\(\sqrt{x-3} \leq 0\)[/tex]
4. [tex]\(x-3 \leq 0\)[/tex]
The appropriate choice is [tex]\(\boxed{x-3 \geq 0}\)[/tex].
