To find \((f - g)(x)\), we start by writing down the expressions for \(f(x)\) and \(g(x)\):
\[ f(x) = \frac{2x + 6}{3x} \]
\[ g(x) = \frac{\sqrt{x} - 8}{3x} \]
Now, we can compute \((f - g)(x)\):
\[ (f - g)(x) = f(x) - g(x) = \frac{2x + 6}{3x} - \frac{\sqrt{x} - 8}{3x} \]
Since they have a common denominator, we can combine the fractions:
\[ (f - g)(x) = \frac{(2x + 6) - (\sqrt{x} - 8)}{3x} \]
Next, we simplify the numerator:
\[ (2x + 6) - (\sqrt{x} - 8) = 2x + 6 - \sqrt{x} + 8 = 2x - \sqrt{x} + 14 \]
Now substituting this back into the fraction gives:
\[ (f - g)(x) = \frac{2x - \sqrt{x} + 14}{3x} \]
Thus, the final result is:
\[ (f - g)(x) = \frac{2x - \sqrt{x} + 14}{3x} \]
It matches the first response option:
\[ \boxed{\frac{2x - \sqrt{x} + 14}{3x}} \]
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