Answer :
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. This is because the square root function is only defined for non-negative values in the context of real numbers.
Here are the detailed steps to find the appropriate inequality:
1. Recognize that for [tex]\( f(x) \)[/tex] to be defined, the expression inside the square root, [tex]\( x - 3 \)[/tex], must be greater than or equal to zero. This ensures that we avoid taking the square root of a negative number, which is not defined in real numbers.
2. Set up the inequality:
[tex]\[ x - 3 \geq 0 \][/tex]
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[ x - 3 \geq 0 \implies x \geq 3 \][/tex]
Therefore, the inequality [tex]\( x - 3 \geq 0 \)[/tex] can be used to find the domain of [tex]\( f(x) = \sqrt{x-3} \)[/tex]. This ensures that [tex]\( f(x) \)[/tex] is defined for all [tex]\( x \)[/tex] values that are greater than or equal to 3.
Hence, the correct inequality is:
[tex]\[ x - 3 \geq 0 \][/tex]
Here are the detailed steps to find the appropriate inequality:
1. Recognize that for [tex]\( f(x) \)[/tex] to be defined, the expression inside the square root, [tex]\( x - 3 \)[/tex], must be greater than or equal to zero. This ensures that we avoid taking the square root of a negative number, which is not defined in real numbers.
2. Set up the inequality:
[tex]\[ x - 3 \geq 0 \][/tex]
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[ x - 3 \geq 0 \implies x \geq 3 \][/tex]
Therefore, the inequality [tex]\( x - 3 \geq 0 \)[/tex] can be used to find the domain of [tex]\( f(x) = \sqrt{x-3} \)[/tex]. This ensures that [tex]\( f(x) \)[/tex] is defined for all [tex]\( x \)[/tex] values that are greater than or equal to 3.
Hence, the correct inequality is:
[tex]\[ x - 3 \geq 0 \][/tex]
