You have the function \( f(x) = \frac{x - 3}{9x + 4} \).

Let's break down some key properties of this function:

### 1. **Domain:**
   The domain of the function includes all values of \( x \) except where the denominator is zero, since division by zero is undefined.

   Set the denominator \( 9x + 4 = 0 \) and solve for \( x \):
   \[
   9x + 4 = 0 \quad \Rightarrow \quad 9x = -4 \quad \Rightarrow \quad x = -\frac{4}{9}
   \]
   Therefore, the domain is:
   \[
   x \in (-\infty, -\frac{4}{9}) \cup (-\frac{4}{9}, \infty)
   \]

### 2. **Vertical Asymptote:**
   A vertical asymptote occurs where the denominator is zero. From the domain, we know there is a vertical asymptote at \( x = -\frac{4}{9} \).

### 3. **Horizontal Asymptote:**
   For rational functions like this one, we can determine the horizontal asymptote by comparing the degrees of the numerator and denominator.
   - The degree of the numerator \( x - 3 \) is 1.
   - The degree of the denominator \( 9x + 4 \) is also 1.

   When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. In this case:
   \[
   \text{Horizontal Asymptote: } y = \frac{1}{9}
   \]

### 4. **Intercepts:**
   - **x-intercept:** Set \( f(x) = 0 \), which happens when the numerator is zero:
     \[
     x - 3 = 0 \quad \Rightarrow \quad x = 3
     \]
     So, the x-intercept is \( (3, 0) \).
   - **y-intercept:** Set \( x = 0 \) and evaluate \( f(0) \):
     \[
     f(0) = \frac{0 - 3}{9(0) + 4} = \frac{-3}{4}
     \]
     So, the y-intercept is \( \left( 0, -\frac{3}{4} \right) \).

Would you like further clarification or additional details on any part of this explanation?

Here are 5 questions to deepen your understanding:
1. How would the function change if the numerator were \( x + 3 \)?
2. What would the domain and asymptotes be for \( f(x) = \frac{2x - 5}{5x + 1} \)?
3. How does the horizontal asymptote affect the behavior of \( f(x) \) as \( x \to \infty \)?
4. Can you find the inverse of this function?
5. What happens to the function's behavior near the vertical asymptote?

**Tip:** Always check for points where the denominator equals zero to avoid undefined values.

Explore the rational function f(x) = (x - 3) / (9x + 4) in detail. Learn how to determine the domain, find vertical and horizontal asymptotes, and calculate intercepts with step-by-step explanations. Suitable for students in Grades 10-12.

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