Question

Question
Determine the domain of extended-left-parenthesis Start Fraction f over g End Fraction extended-right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals negative Start Fraction 1 over x End Fraction and lower g left-parenthesis x right-parenthesis equals Start Root 3 x minus 9 End Root.
(1 point)
Responses

left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis 3 comma infinity right-parenthesis

left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis

left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis

left-parenthesis negative infinity comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
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Answer 1

To determine the domain of the function \(\frac{f}{g}(x)\) where \(f(x) = -\frac{1}{x}\) and \(g(x) = \sqrt{3x - 9}\), we need to find values of \(x\) for which both \(f(x)\) and \(g(x)\) are defined and \(g(x) \neq 0\).

1. **For \(f(x) = -\frac{1}{x}\):**
- \(f(x)\) is defined for all \(x\) except \(x = 0\) (since division by zero is undefined).
- Therefore, the condition for \(f(x)\) is \(x \neq 0\).

2. **For \(g(x) = \sqrt{3x - 9}\):**
- The expression inside the square root must be non-negative:
\[
3x - 9 \geq 0
\]
\[
3x \geq 9
\]
\[
x \geq 3
\]
- Additionally, \(g(x)\) must be greater than zero, since we cannot have \(g(x) = 0\) in the denominator of the fraction:
\[
\sqrt{3x - 9} > 0 \implies 3x - 9 > 0 \implies 3x > 9 \implies x > 3
\]

So, for \(g(x)\) to be defined and non-zero, we require \(x > 3\).

3. **Combining Conditions:**
- We have from \(f(x)\) that \(x \neq 0\) and from \(g(x)\) that \(x > 3\).
- Therefore, the domain of \(\frac{f}{g}(x)\) is \(x > 3\).

4. **Final Domain:**
- The domain can be expressed in interval notation as:
\[
(3, \infty)
\]

Thus, the answer is \((3, \infty)\).