It looks like you're trying to define the function \( F(x) = \frac{|x - 2|}{x - 4} \). This is a rational function where the numerator involves the absolute value of \( x - 2 \), and the denominator is a linear expression \( x - 4 \).

### Key Points to Consider:
1. **Domain**: The function is undefined when the denominator is zero, which occurs at \( x = 4 \). Therefore, the domain of \( F(x) \) is all real numbers except \( x = 4 \).
  
2. **Behavior of the Absolute Value**:
   - For \( x \geq 2 \), \( |x - 2| = x - 2 \).
   - For \( x < 2 \), \( |x - 2| = 2 - x \).

3. **Piecewise Definition**:
   The function can be rewritten in piecewise form:
   \[
   F(x) =
   \begin{cases}
   \frac{x - 2}{x - 4} & \text{if } x \geq 2 \text{ and } x \neq 4, \\
   \frac{2 - x}{x - 4} & \text{if } x < 2.
   \end{cases}
   \]

4. **Asymptotic Behavior**:
   - **Vertical Asymptote**: At \( x = 4 \), the function has a vertical asymptote because the denominator goes to zero, leading to an undefined expression.
   - **End Behavior**: As \( x \) approaches \( \pm\infty \), \( F(x) \) will tend to 1 (if \( x > 2 \)) or -1 (if \( x < 2 \)), respectively.

Would you like to explore more details, such as graphing this function or analyzing specific points?

### Related Questions:
1. How do you graph the piecewise function \( F(x) = \frac{|x - 2|}{x - 4} \)?
2. What is the behavior of \( F(x) \) near the vertical asymptote at \( x = 4 \)?
3. How do you determine the limits of \( F(x) \) as \( x \) approaches 4 from the left and right?
4. What is the domain and range of \( F(x) = \frac{|x - 2|}{x - 4} \)?
5. How does the absolute value affect the shape of the graph for \( F(x) \)?

### Tip:
Always check for points where the function might be undefined, such as when the denominator is zero, to understand the complete behavior of the function.

Explore the rational function F(x) = |x - 2| / (x - 4), its domain, piecewise definition, asymptotic behavior, and more. This detailed analysis covers concepts like rational functions, absolute value, and asymptotes, suitable for advanced high school students.

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