The function you provided is:

\[
F(x) = \frac{1}{x - 2} + 3
\]

This is a rational function where the variable \(x\) appears in the denominator of the first term. Let's break it down:

1. The term \(\frac{1}{x - 2}\) represents a rational function with a vertical asymptote at \(x = 2\) (because the denominator becomes zero at \(x = 2\), which makes the function undefined).
2. The constant \(+3\) shifts the entire graph of the function upwards by 3 units.

### Key features of the function:
- **Vertical asymptote**: at \(x = 2\).
- **Horizontal asymptote**: Since the constant term is \(+3\), the horizontal asymptote is \(y = 3\), because as \(x\) becomes very large or very small, \(\frac{1}{x - 2}\) approaches 0.
- **Domain**: All real numbers except \(x = 2\), i.e., \( (-\infty, 2) \cup (2, \infty) \).

Would you like to go into further details or clarify any part of this?

Here are 5 related questions to expand this concept:
1. How would the graph change if we altered the constant 3 to another number?
2. What is the behavior of the function as \(x \to 2\) from the left and right sides?
3. Can we calculate the derivative of this function?
4. How would you find the x-intercept of this function?
5. What is the effect of modifying the denominator to \(x - a\) instead of \(x - 2\)?

**Tip**: For rational functions like these, always identify vertical and horizontal asymptotes, as they provide key information about the function's behavior.

Explore the rational function F(x) = 1/(x-2) + 3, including its vertical and horizontal asymptotes, domain, and graph behavior. Learn how to analyze key features of the function and apply mathematical concepts for students in Grades 9-11.

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