Let's analyze the given function and identify its features step-by-step.

The given function is:

$f(x) = \frac{-6x - 9}{4x}$

Step 1: Simplify the function

Factorize the numerator: $-6x - 9 = -3(2x + 3)$ So, $f(x) = \frac{-3(2x + 3)}{4x}$

Step 2: Find the Vertical Asymptote

Vertical asymptotes occur where the denominator is zero but the numerator is not zero.

The denominator is $4x$. Setting it to zero: $4x = 0 \implies x = 0$

Thus, there is a vertical asymptote at $x = 0$.

Step 3: Find the Horizontal Asymptote

Horizontal asymptotes depend on the degree of the numerator and denominator.

• The degree of the numerator ($-6x - 9$) is 1.
• The degree of the denominator ($4x$) is also 1.

For rational functions where the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients: $\text{Horizontal Asymptote: } y = \frac{-6}{4} = -\frac{3}{2}$

Thus, there is a horizontal asymptote at $y = -\frac{3}{2}$.

Step 4: Find the $x$-Intercept

To find the $x$-intercept, set $f(x) = 0$. This happens when the numerator is zero: $-6x - 9 = 0 \implies -6x = 9 \implies x = -\frac{3}{2}$

Thus, the $x$-intercept is at $(-3/2, 0)$.

Step 5: Find the $y$-Intercept

To find the $y$-intercept, set $x = 0$ in the function. However, since the denominator is $4x$, the function is undefined at $x = 0$. Therefore, there is no $y$-intercept.

Step 6: Identify Holes

Holes occur when a factor in the numerator cancels with a factor in the denominator. There are no common factors between the numerator ($-6x - 9$) and denominator ($4x$). Thus, there are no holes.

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Final Answers:

1. Horizontal Asymptote: $y = -\frac{3}{2}$
2. Vertical Asymptote: $x = 0$
3. $x$-Intercept: $(-3/2, 0)$
4. $y$-Intercept: None
5. Holes: None

Would you like further clarification or a graph of the function? Here are five related questions to deepen your understanding:

1. How do you determine asymptotes for rational functions in general?
2. What happens if a hole exists in the function?
3. Why is there no $y$-intercept in this case?
4. How would you graph the function based on these features?
5. What is the domain of the function?

Tip: Always simplify the function before identifying its features—it makes the analysis clearer!