The function you've provided is:

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

This is a rational function composed of two parts:
1. \(-\frac{2}{x}\) — a rational term with an asymptote at \(x = 0\).
2. \(+3\) — a constant term that shifts the entire function vertically.

### Key properties of this function:

1. **Domain**: The function is undefined where the denominator is zero. Therefore, the domain is all real numbers except \(x = 0\):
   \[
   x \neq 0
   \]
2. **Vertical Asymptote**: Since the term \(-\frac{2}{x}\) becomes undefined when \(x = 0\), there is a vertical asymptote at \(x = 0\).
   
3. **Horizontal Asymptote**: As \(x\) approaches infinity (either positive or negative), \(-\frac{2}{x}\) approaches 0. Therefore, the horizontal asymptote is determined by the constant term \(+3\). So, the horizontal asymptote is:
   \[
   y = 3
   \]

4. **Intercepts**:
   - **Y-intercept**: To find the y-intercept, set \(x = 0\), but since \(x = 0\) is excluded from the domain, there is no y-intercept.
   - **X-intercept**: Set \(F(x) = 0\) and solve for \(x\):
     \[
     0 = -\frac{2}{x} + 3
     \]
     \[
     \frac{2}{x} = 3
     \]
     \[
     x = \frac{2}{3}
     \]
     So, the x-intercept is at \(x = \frac{2}{3}\).

Would you like to see a graph of this function or have further details?

Here are some related questions to think about:
1. What is the behavior of the function as \(x \to 0^+\) and \(x \to 0^-\)?
2. How does the function behave for very large positive and negative \(x\)?
3. What happens to the function if we change the coefficient of \(\frac{2}{x}\)?
4. What is the effect of changing the constant term from \(3\) to another number?
5. Can we express the function in terms of transformations of a simpler function?

**Tip**: Understanding the asymptotes can help you sketch rational functions quickly!

Explore the rational function F(x) = -2/x + 3, including its vertical and horizontal asymptotes, domain, and intercepts. Learn key mathematical concepts such as domain restrictions, behavior near asymptotes, and how to solve for intercepts. Ideal for students in Grades 10-12.

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