Given the function \( f(x) = \frac{x-2}{4x - x^2} \), we can analyze it as follows:

### 1. **Simplification:**
   - First, let's factor the denominator:
   \[
   4x - x^2 = x(4 - x)
   \]
   So, the function can be rewritten as:
   \[
   f(x) = \frac{x-2}{x(4-x)}
   \]

### 2. **Domain:**
   - The function is undefined where the denominator equals zero. Setting the denominator to zero:
   \[
   x(4-x) = 0
   \]
   This gives us:
   \[
   x = 0 \quad \text{or} \quad x = 4
   \]
   So, the domain of \( f(x) \) is \( x \in \mathbb{R} \) except \( x = 0 \) and \( x = 4 \).

### 3. **Vertical Asymptotes:**
   - Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Therefore, vertical asymptotes are at:
   \[
   x = 0 \quad \text{and} \quad x = 4
   \]

### 4. **Horizontal Asymptote:**
   - To find the horizontal asymptote, observe the behavior of \( f(x) \) as \( x \) approaches infinity. The highest degree terms in the numerator and denominator are:
   \[
   f(x) \approx \frac{-x^2}{-x^2} = \frac{1}{1} = 1
   \]
   Hence, the horizontal asymptote is \( y = 1 \).

### 5. **Intercepts:**
   - **X-intercept:** Set \( f(x) = 0 \):
   \[
   \frac{x-2}{x(4-x)} = 0
   \]
   The numerator must be zero:
   \[
   x - 2 = 0 \quad \Rightarrow \quad x = 2
   \]
   So, the x-intercept is at \( (2, 0) \).
   - **Y-intercept:** Set \( x = 0 \) in \( f(x) \):
   \[
   f(0) \text{ is undefined} \quad (\text{since \( x = 0 \) is not in the domain})
   \]

Would you like more details on any part of this analysis, or do you have specific questions?

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**Related Questions:**
1. How do we determine the horizontal asymptote of any rational function?
2. What are the key steps in finding the vertical asymptotes of a function?
3. Can a function have more than one horizontal asymptote?
4. How do we analyze the end behavior of a rational function?
5. What happens to the graph of the function near the vertical asymptotes?

**Tip:** When simplifying a rational function, always factor the numerator and denominator first to identify common factors or potential cancellations.

Explore the rational function f(x) = (x - 2) / (4x - x^2) with a detailed analysis of its domain, vertical and horizontal asymptotes, and intercepts. This solution is perfect for students in Grades 10-12, covering key concepts in algebra and rational function behavior.

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