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 under the square root is non-negative. The square root function is only defined for non-negative arguments.
1. Step 1: Identify the condition for the square root to be defined.
The function [tex]\( f(x) = \sqrt{x-2} \)[/tex] will be defined if the expression inside the square root, [tex]\( x-2 \)[/tex], is greater than or equal to zero:
[tex]\[
x - 2 \geq 0
\][/tex]
2. Step 2: Solve the inequality.
To solve the inequality [tex]\( x - 2 \geq 0 \)[/tex]:
[tex]\[
x \geq 2
\][/tex]
Therefore, the domain of [tex]\( f(x) = \sqrt{x-2} \)[/tex] is all values of [tex]\( x \)[/tex] such that [tex]\( x \geq 2 \)[/tex].
3. Step 3: Check each given option.
- Option A: [tex]\( x = 1 \)[/tex]
[tex]\[
1 \geq 2 \quad \text{is false}
\][/tex]
So, [tex]\( x = 1 \)[/tex] is not in the domain.
- Option B: [tex]\( x = -2 \)[/tex]
[tex]\[
-2 \geq 2 \quad \text{is false}
\][/tex]
So, [tex]\( x = -2 \)[/tex] is not in the domain.
- Option C: [tex]\( x = 2 \)[/tex]
[tex]\[
2 \geq 2 \quad \text{is true}
\][/tex]
So, [tex]\( x = 2 \)[/tex] is in the domain.
- Option D: [tex]\( x = 0 \)[/tex]
[tex]\[
0 \geq 2 \quad \text{is false}
\][/tex]
So, [tex]\( x = 0 \)[/tex] is not in the domain.
4. Conclusion:
The only value among the given options that satisfies the inequality [tex]\( x \geq 2 \)[/tex] 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]\[ \boxed{x=2} \][/tex]
