Which value of [tex][tex]$x$[/tex][/tex] is in the domain of [tex][tex]$f(x) = \sqrt{x-2}$[/tex][/tex]?

A. [tex][tex]$x = 1$[/tex][/tex]

B. [tex][tex]$x = -2$[/tex][/tex]

C. [tex][tex]$x = 2$[/tex][/tex]

D. [tex][tex]$x = 0$[/tex][/tex]



Answer :

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]