The problem involves determining the correct rational function $f(x)$ given its graphical properties and the options provided. Let's break down the important information and use it to construct the equation.

Key Observations:

1. Vertical Asymptotes: The vertical asymptotes are at $x = -1$ and $x = -6$.

   • This indicates that the denominator of the rational function has factors corresponding to $x + 1$ and $x + 6$, since vertical asymptotes occur where the denominator equals zero.

2. Horizontal Asymptote: The horizontal asymptote is $y = 0$.

   • A horizontal asymptote of $y = 0$ suggests that the degree of the numerator is less than the degree of the denominator.

3. x-Intercept: The graph has an x-intercept at $x = -2$.

   • This means $f(-2) = 0$, so the numerator must have a factor of $x + 2$.

4. Point on the Graph: The graph passes through $(-4, -1)$.

   • This gives us a specific point to use for finding the value of any constant in the equation.

Choosing the Equation Form:

We know the vertical asymptotes must be represented by factors in the denominator. The most likely form for the function is: $f(x) = \frac{a(x + 2)}{(x + 1)(x + 6)}$ This corresponds to the third option on the list, which matches the asymptotes and the x-intercept.

Finding the Constant $a$:

Now, we will use the point $(-4, -1)$ to solve for $a$.

Substitute $x = -4$ and $f(x) = -1$ into the equation: $-1 = \frac{a(-4 + 2)}{(-4 + 1)(-4 + 6)}$ Simplifying: $-1 = \frac{a(-2)}{(-3)(2)}$ $-1 = \frac{-2a}{-6} = \frac{a}{3}$ Multiplying both sides by 3: $a = -3$

Final Equation:

Thus, the equation of the rational function is: $f(x) = \frac{-3(x + 2)}{(x + 1)(x + 6)}$

Let me know if you need more details or further explanation.

Here are five related questions to deepen your understanding:

1. What effect does changing the sign of $a$ have on the graph of a rational function?
2. How do we identify vertical and horizontal asymptotes from the equation of a rational function?
3. Can a rational function have more than two vertical asymptotes? If so, how is that reflected in the equation?
4. Why does the degree of the numerator being less than the degree of the denominator result in a horizontal asymptote at $y = 0$?
5. How can we find the y-intercept of a rational function like the one above?

Tip: Always use known points from the graph to solve for unknown constants in the function's equation.