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.