Questions: Find the domain of the following rational function F(x) = 9 x(x-4) / (6 x^2 - 41 x - 7) Select the correct choice below and, if necessary, fill in any answer boxes within your choice. A. The domain of F(x) is (s ? (Type in inequality in the form a<x<b. Use integers or fractions for any numbers in the expression. Use a comma to separate entries as needed.) B. The domain of F(x) is the set of all real numbers.

Solution Steps

To find the domain of the function \( F(x) = \frac{9x(x-4)}{6x^2 - 41x - 7} \), we need to determine the values of \( x \) for which the denominator is not zero, as division by zero is undefined. This involves solving the quadratic equation \( 6x^2 - 41x - 7 = 0 \) to find the values of \( x \) that make the denominator zero. The domain will be all real numbers except these values.

To find the domain of the function \( F(x) = \frac{9x(x-4)}{6x^2 - 41x - 7} \), we need to determine the values of \( x \) for which the denominator is zero. The denominator is given by: \[ 6x^2 - 41x - 7 \]

We solve the quadratic equation \( 6x^2 - 41x - 7 = 0 \) to find the values of \( x \) that make the denominator zero. The solutions to this equation are: \[ x = -\frac{1}{6} \quad \text{and} \quad x = 7 \]

The domain of \( F(x) \) is all real numbers except the values that make the denominator zero. Therefore, the domain is: \[ x \in \mathbb{R} \setminus \left\{ -\frac{1}{6}, 7 \right\} \]

Final Answer