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The function provided is:

\[
f(x) = \frac{4(x - 1)}{3(x - 1)^3}
\]

We can simplify this function and analyze its features step by step.

### Step 1: Simplification
Simplifying the given function:

\[
f(x) = \frac{4(x - 1)}{3(x - 1)^3} = \frac{4}{3(x - 1)^2}
\]

So the function simplifies to:

\[
f(x) = \frac{4}{3(x - 1)^2}
\]

### Step 2: Identifying key features

1. **Domain**: The function has a denominator of \( (x - 1)^2 \), meaning the function is undefined when \( x = 1 \). Therefore, the domain is all real numbers except \( x = 1 \).
   - Domain: \( (-\infty, 1) \cup (1, \infty) \)

2. **Vertical Asymptote**: Since the function is undefined at \( x = 1 \), and the denominator approaches zero as \( x \) approaches 1, there is a vertical asymptote at \( x = 1 \).

3. **Horizontal Asymptote**: As \( x \to \pm\infty \), the denominator \( (x - 1)^2 \) grows very large, so the function tends to zero. Therefore, there is a horizontal asymptote at \( y = 0 \).

4. **Behavior near the Asymptote (x = 1)**: As \( x \) approaches 1 from the left (\( x \to 1^- \)) and the right (\( x \to 1^+ \)), the function tends to infinity because of the \( (x - 1)^2 \) in the denominator. So, both sides of the vertical asymptote rise towards infinity.

5. **Symmetry**: The function involves only \( (x - 1)^2 \), which is an even function. This indicates symmetry about \( x = 1 \). Therefore, the graph is symmetric around the vertical asymptote.

### Summary of Features:
- **Domain**: \( (-\infty, 1) \cup (1, \infty) \)
- **Vertical Asymptote**: \( x = 1 \)
- **Horizontal Asymptote**: \( y = 0 \)
- **Symmetry**: Symmetric about \( x = 1 \)
- **Behavior near Asymptote**: Tends to infinity as \( x \to 1^+ \) and \( x \to 1^- \)

Would you like to go over any specific details, or do you have any questions?

Here are five related questions to explore this further:
1. How would the graph of \( f(x) \) change if we replaced \( (x-1)^2 \) with \( (x-1)^3 \)?
2. How can we find the intervals of increase or decrease for this function?
3. What is the second derivative of \( f(x) \) and what does it tell us about concavity?
4. What happens to the function \( f(x) \) when \( x \to 0 \) or \( x \to 2 \)?
5. How does changing the constant coefficients (e.g., 4 and 3) affect the asymptotes?

**Tip**: When finding the behavior near an asymptote, always check both directions for vertical and horizontal tendencies.

Determine each feature of the graph of the given function f(x) = 4(x - 1) / 3(x - 1)^3.

Analyze the graph features of the rational function f(x) = 4(x - 1) / 3(x - 1)^3. Learn how to determine the domain, vertical and horizontal asymptotes, and symmetry. This problem provides a detailed step-by-step simplification and analysis, ideal for students in Grades 10-12.

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