To determine which value of [tex]\( x \)[/tex] is in the domain of the function [tex]\( f(x) = \sqrt{x-8} \)[/tex], we need to ensure that the expression inside the square root is non-negative. This means we need:
[tex]\[ x - 8 \geq 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x \geq 8 \][/tex]
Let's evaluate each provided value of [tex]\( x \)[/tex] to see if it satisfies this inequality:
1. Option A: [tex]\( x = 10 \)[/tex]
- Substituting [tex]\( 10 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ 10 - 8 = 2 \][/tex]
- Since [tex]\( 2 \geq 0 \)[/tex], [tex]\( x = 10 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
2. Option B: [tex]\( x = -8 \)[/tex]
- Substituting [tex]\( -8 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ -8 - 8 = -16 \][/tex]
- Since [tex]\( -16 \)[/tex] is not greater than or equal to [tex]\( 0 \)[/tex], [tex]\( x = -8 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
3. Option C: [tex]\( x = 0 \)[/tex]
- Substituting [tex]\( 0 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ 0 - 8 = -8 \][/tex]
- Since [tex]\( -8 \)[/tex] is not greater than or equal to [tex]\( 0 \)[/tex], [tex]\( x = 0 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
4. Option D: [tex]\( x = 7 \)[/tex]
- Substituting [tex]\( 7 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ 7 - 8 = -1 \][/tex]
- Since [tex]\( -1 \)[/tex] is not greater than or equal to [tex]\( 0 \)[/tex], [tex]\( x = 7 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
Therefore, the only value of [tex]\( x \)[/tex] that is in the domain of [tex]\( f(x) = \sqrt{x-8} \)[/tex] is:
A. [tex]\( x = 10 \)[/tex]
