To find the domain of the rational function $F(x) = \frac{4x(x - 3)}{2x^2 - 9x - 5}$, we need to determine when the denominator is zero, as a rational function is undefined at these points.

Step 1: Factor the Denominator

The denominator is $2x^2 - 9x - 5$. Let's factor this expression to find values of $x$ that make it zero: $2x^2 - 9x - 5 = (2x + 1)(x - 5)$

Step 2: Solve for the Zeroes

Now, we set each factor equal to zero to find the values of $x$ that make the denominator zero:

1. $2x + 1 = 0 \Rightarrow x = -\frac{1}{2}$
2. $x - 5 = 0 \Rightarrow x = 5$

Step 3: Determine the Domain

Since $x = -\frac{1}{2}$ and $x = 5$ make the denominator zero, these values must be excluded from the domain. Therefore, the domain of $F(x)$ is all real numbers except $x = -\frac{1}{2}$ and $x = 5$.

Answer

The correct choice is: A. The domain of $F(x)$ is $\{ x \mid x \neq -\frac{1}{2}, 5 \}$.

Would you like further details or explanations on factoring or determining the domain?

Related Questions

1. How do you find the domain of a general rational function?
2. What is the significance of factors in the denominator of a rational function?
3. How can we identify asymptotes from the domain restrictions?
4. What methods exist for factoring quadratic expressions?
5. How do vertical asymptotes relate to the domain of a function?

Tip

For rational functions, always start by factoring the denominator to identify restrictions in the domain.