To determine the domain of the function \(\frac{f}{g}(x) = \frac{-\frac{1}{x}}{\sqrt{3x - 9}}\), you need to analyze both the numerator and the denominator:
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Numerator \(f(x) = -\frac{1}{x}\):
- This function is defined for all \(x\) except \(x = 0\). Thus, \(x \neq 0\).
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Denominator \(g(x) = \sqrt{3x - 9}\):
- The square root function is defined for non-negative values, so we need:
\[3x - 9 \geq 0\]
\[3x \geq 9\]
\[x \geq 3\] - Also, we must ensure that the denominator is not zero, which occurs when \(3x - 9 = 0\) or \(x = 3\). Hence, \(x\) cannot equal \(3\).
- The square root function is defined for non-negative values, so we need:
So, combining these constraints, we know that \(x\) must be greater than or equal to 3 while also excluding the point \(3\). Therefore, the domain of \(\frac{f}{g}(x)\) is:
\[ (3, \infty) \]
Thus, the correct response is: \[ \left(3, \infty\right) \]