To determine which function correctly defines [tex]\((f-g)(x)\)[/tex], we need to find the difference between the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Given:
[tex]\[ f(x) = \sqrt{\frac{x}{8}} + 11 \][/tex]
[tex]\[ g(x) = 5 + \frac{2}{x} \][/tex]
We compute [tex]\((f-g)(x)\)[/tex] by subtracting [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]:
[tex]\[
(f-g)(x) = f(x) - g(x)
\][/tex]
Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f-g)(x) = \left( \sqrt{\frac{x}{8}} + 11 \right) - \left( 5 + \frac{2}{x} \right)
\][/tex]
Simplify the expression:
[tex]\[
(f-g)(x) = \sqrt{\frac{x}{8}} + 11 - 5 - \frac{2}{x}
\][/tex]
Combine the constant terms:
[tex]\[
(f-g)(x) = \sqrt{\frac{x}{8}} + 6 - \frac{2}{x}
\][/tex]
Now we compare this simplified form of [tex]\( (f-g)(x) \)[/tex] with the given choices:
A. [tex]\( (f-g)(x) = \sqrt{\frac{x}{8}} + \frac{2}{x} - 6 \)[/tex]
B. [tex]\( (f-g)(x) = \sqrt{\frac{x}{8}} - \frac{2}{x} + 6 \)[/tex]
C. [tex]\( (f-g)(x) = \sqrt{\frac{x}{8} - \frac{2}{x}} + 16 \)[/tex]
D. [tex]\( (f-g)(x) = \sqrt{\frac{x}{8}} + \frac{2}{x} - 16 \)[/tex]
The correct expression must match our simplified form [tex]\( \sqrt{\frac{x}{8}} + 6 - \frac{2}{x} \)[/tex].
Comparing this form with the choices, we see:
- Choice B: [tex]\( \sqrt{\frac{x}{8}} - \frac{2}{x} + 6 \)[/tex] matches exactly.
Thus, the correct choice is:
B. [tex]\( (f-g)(x) = \sqrt{\frac{x}{8}} - \frac{2}{x} + 6 \)[/tex].
