To determine which value of [tex]\( x \)[/tex] is in the domain of the function [tex]\( f(x) = \sqrt{x-2} \)[/tex], we need to ensure that the expression inside the square root is non-negative. Square roots are only defined for non-negative values. Thus, the expression [tex]\( x-2 \)[/tex] must be greater than or equal to zero.
Let's solve the inequality step-by-step:
1. Write the inequality:
[tex]\[
x - 2 \geq 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq 2
\][/tex]
This means that for [tex]\( f(x) \)[/tex] to be defined, [tex]\( x \)[/tex] must be at least 2 or greater.
Now, let's evaluate each of the provided choices:
- A. [tex]\( x = -2 \)[/tex]:
[tex]\[
-2 - 2 = -4 \quad (\text{which is less than } 0)
\][/tex]
So, [tex]\( \sqrt{-4} \)[/tex] is not defined.
- B. [tex]\( x = 0 \)[/tex]:
[tex]\[
0 - 2 = -2 \quad (\text{which is less than } 0)
\][/tex]
So, [tex]\( \sqrt{-2} \)[/tex] is not defined.
- C. [tex]\( x = 1 \)[/tex]:
[tex]\[
1 - 2 = -1 \quad (\text{which is less than } 0)
\][/tex]
So, [tex]\( \sqrt{-1} \)[/tex] is not defined.
- D. [tex]\( x = 2 \)[/tex]:
[tex]\[
2 - 2 = 0 \quad (\text{which is equal to } 0)
\][/tex]
So, [tex]\( \sqrt{0} \)[/tex] is defined and equal to 0.
Given the inequality [tex]\( x \geq 2 \)[/tex], the only choice that satisfies this condition is [tex]\( x = 2 \)[/tex].
Therefore, the value of [tex]\( x \)[/tex] that is in the domain of [tex]\( f(x) = \sqrt{x-2} \)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
