Given the function \( f(x) = \frac{x - 2}{4x - x^2} \), let's analyze it in more detail, including its domain, intercepts, asymptotes, and general behavior.

### 1. **Domain**:
The domain of the function consists of all values of \( x \) for which the function is defined. This requires that the denominator does not equal zero.

The denominator is \( 4x - x^2 \), which can be factored as:
\[
4x - x^2 = x(4 - x)
\]
Setting this equal to zero gives:
\[
x(4 - x) = 0
\]
So, \( x = 0 \) or \( x = 4 \).

Therefore, the function is undefined at \( x = 0 \) and \( x = 4 \). Thus, the domain of \( f(x) \) is:
\[
x \in (-\infty, 0) \cup (0, 4) \cup (4, \infty)
\]

### 2. **Intercepts**:
- **Y-intercept**: To find the y-intercept, set \( x = 0 \). However, as mentioned, the function is undefined at \( x = 0 \), so there is no y-intercept.

- **X-intercept**: To find the x-intercept, set \( f(x) = 0 \). This implies the numerator should equal zero:
\[
x - 2 = 0
\]
So, \( x = 2 \). The x-intercept is \( (2, 0) \).

### 3. **Asymptotes**:
- **Vertical Asymptotes**: These occur where the denominator equals zero (and the function is undefined).
  - From the domain analysis, vertical asymptotes are at \( x = 0 \) and \( x = 4 \).

- **Horizontal Asymptote**: For large values of \( x \), the behavior of the function can be determined by comparing the degrees of the numerator and denominator.
  - The degree of the numerator (linear) is 1, and the degree of the denominator (quadratic) is 2.
  - Since the degree of the denominator is higher, the horizontal asymptote is \( y = 0 \).

### 4. **General Behavior**:
- **For \( x \to -\infty \)** and **\( x \to \infty \)**, the function approaches the horizontal asymptote \( y = 0 \).
- **At \( x = 2 \)**, the function crosses the x-axis.

Would you like to explore more details on the graphing, limits, or other properties of this function?

### Relative Questions:
1. What happens to the function's behavior near the vertical asymptotes \( x = 0 \) and \( x = 4 \)?
2. How can you determine if there are any turning points in the function \( f(x) \)?
3. Can the function have any oblique asymptotes? Why or why not?
4. What would be the result of taking the derivative of this function to find critical points?
5. How do you find the exact value of the function as \( x \) approaches 4 from either side?

### Tip:
When analyzing the behavior of a rational function near asymptotes, checking limits from both sides of the asymptote helps understand how the function behaves near undefined points.

Explore the detailed analysis of the rational function f(x) = (x - 2) / (4x - x^2), including domain, intercepts, vertical and horizontal asymptotes. This problem is suitable for students in Grades 11-12, covering key concepts in rational functions.

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