Answer :
To determine the domain of the function [tex]\(f(x) = \sqrt{x-3}\)[/tex], we must understand the properties of the square root function. The square root function is defined only for non-negative numbers because the square root of a negative number is not a real number.
To ensure that the expression inside the square root, [tex]\(x-3\)[/tex], is non-negative, we must solve the following inequality:
[tex]\[ x - 3 \geq 0 \][/tex]
This inequality ensures that the argument of the square root is zero or positive, which is required for the square root function to produce a real number.
Solving the inequality [tex]\(x - 3 \geq 0\)[/tex]:
1. Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x \geq 3 \][/tex]
Thus, the inequality that can be used to find the domain of [tex]\(f(x) = \sqrt{x-3}\)[/tex] is:
[tex]\[ x - 3 \geq 0 \][/tex]
So, the correct answer is:
[tex]\[ x-3 \geq 0 \][/tex]
To ensure that the expression inside the square root, [tex]\(x-3\)[/tex], is non-negative, we must solve the following inequality:
[tex]\[ x - 3 \geq 0 \][/tex]
This inequality ensures that the argument of the square root is zero or positive, which is required for the square root function to produce a real number.
Solving the inequality [tex]\(x - 3 \geq 0\)[/tex]:
1. Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x \geq 3 \][/tex]
Thus, the inequality that can be used to find the domain of [tex]\(f(x) = \sqrt{x-3}\)[/tex] is:
[tex]\[ x - 3 \geq 0 \][/tex]
So, the correct answer is:
[tex]\[ x-3 \geq 0 \][/tex]
