To determine which values of [tex]\( x \)[/tex] are in the domain of the function [tex]\( f(x) = \sqrt{x-2} \)[/tex], we need to consider the expression inside the square root. The square root function is defined only for non-negative values. Therefore, the expression under the square root must be greater than or equal to zero.
Let's analyze the inequality:
[tex]\[ x - 2 \geq 0 \][/tex]
Solve this inequality for [tex]\( x \)[/tex]:
1. Add 2 to both sides:
[tex]\[ x \geq 2 \][/tex]
This inequality tells us that [tex]\( x \)[/tex] must be greater than or equal to 2 for the function [tex]\( f(x) = \sqrt{x-2} \)[/tex] to be defined.
Now, let's examine the given options to see which values satisfy this condition:
A. [tex]\( x = -2 \)[/tex]
[tex]\[ -2 - 2 = -4 \][/tex]
The expression [tex]\( -4 \)[/tex] is negative, so [tex]\( x = -2 \)[/tex] is not in the domain.
B. [tex]\( x = 0 \)[/tex]
[tex]\[ 0 - 2 = -2 \][/tex]
The expression [tex]\( -2 \)[/tex] is negative, so [tex]\( x = 0 \)[/tex] is not in the domain.
C. [tex]\( x = 1 \)[/tex]
[tex]\[ 1 - 2 = -1 \][/tex]
The expression [tex]\( -1 \)[/tex] is negative, so [tex]\( x = 1 \)[/tex] is not in the domain.
D. [tex]\( x = 2 \)[/tex]
[tex]\[ 2 - 2 = 0 \][/tex]
The expression [tex]\( 0 \)[/tex] is non-negative, so [tex]\( x = 2 \)[/tex] is in the domain.
Therefore, the smallest value of [tex]\( x \)[/tex] that is in the domain of [tex]\( f(x) = \sqrt{x-2} \)[/tex] is 2.
The correct answer is:
[tex]\[ \boxed{2} \][/tex]
