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
(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
Answers
GPT-4o mini
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:
1. **Numerator \(f(x) = -\frac{1}{x}\)**:
- This function is defined for all \(x\) except \(x = 0\). Thus, \(x \neq 0\).
2. **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 \(3\).
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:
\[
\left(3, \infty\right)
\]
1. **Numerator \(f(x) = -\frac{1}{x}\)**:
- This function is defined for all \(x\) except \(x = 0\). Thus, \(x \neq 0\).
2. **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 \(3\).
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:
\[
\left(3, \infty\right)
\]