Find the domain of the following rational function.
Upper F left parenthesis x right parenthesis equals StartFraction 4 x left parenthesis x minus 6 right parenthesis Over 3 x squared minus 11 x minus 4 EndFraction

To find the domain of the rational function
\[
F(x) = \frac{4x(x - 6)}{3x^2 - 11x - 4},
\]
we need to identify values of \( x \) that make the denominator equal to zero, since division by zero is undefined.

### Step 1: Set the denominator equal to zero
The denominator is:
\[
3x^2 - 11x - 4 = 0
\]
To solve this quadratic equation, we'll use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \(a = 3\), \(b = -11\), and \(c = -4\).

### Step 2: Calculate the discriminant
\[
b^2 - 4ac = (-11)^2 - 4(3)(-4) = 121 + 48 = 169.
\]

### Step 3: Find the roots using the quadratic formula
\[
x = \frac{-(-11) \pm \sqrt{169}}{2(3)},
\]
\[
x = \frac{11 \pm 13}{6}.
\]
This gives us:
\[
x = \frac{24}{6} = 4 \quad \text{and} \quad x = \frac{-2}{6} = -\frac{1}{3}.
\]

### Step 4: Determine the domain
The values \( x = 4 \) and \( x = -\frac{1}{3} \) make the denominator zero, so they must be excluded from the domain.

Therefore, the domain of \( F(x) \) is all real numbers except \( x = 4 \) and \( x = -\frac{1}{3} \).

### Final Answer
The domain of \( F(x) \) is:
\[
\{ x \in \mathbb{R} : x \neq 4 \text{ and } x \neq -\frac{1}{3} \}.
\]

Would you like more details or have any questions?

### 5 Related Questions
1. How do you find the domain of a rational function in general?
2. What happens to the function's behavior as \( x \) approaches 4 or \(-\frac{1}{3}\)?
3. How can we factor a quadratic expression like \(3x^2 - 11x - 4\) without using the quadratic formula?
4. How does the numerator affect the domain of a rational function?
5. How do you determine the vertical asymptotes of a rational function?

### Tip
When finding the domain of a rational function, always start by identifying values that make the denominator zero, as these values are excluded from the domain.

Find the domain of the following rational function. F(x) = (4x(x - 6)) / (3x^2 - 11x - 4)

Discover how to find the domain of the rational function F(x) = (4x(x - 6)) / (3x^2 - 11x - 4). This step-by-step solution uses the quadratic formula to solve for x-values that make the denominator zero, excluding them from the domain. Suitable for students in Grades 9-11.

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