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.