The function $F(x) = \frac{2}{x}$ is a rational function. Its behavior and properties can be analyzed as follows:

Key Points of $F(x) = \frac{2}{x}$:

1. Domain:

   • The function is defined for all real numbers except $x = 0$, since division by zero is undefined.
   • Domain: $x \in \mathbb{R} \setminus \{0\}$.

2. Range:

   • The range includes all real numbers except $F(x) = 0$ because the numerator is nonzero.
   • Range: $F(x) \in \mathbb{R} \setminus \{0\}$.

3. Asymptotes:

   • Vertical Asymptote: $x = 0$ (the function approaches infinity or negative infinity as $x \to 0$ from either side).
   • Horizontal Asymptote: $F(x) = 0$ (the function approaches 0 as $x \to \pm\infty$).

4. Symmetry:

   • The function is odd: $F(-x) = -F(x)$. This symmetry implies that the graph is symmetric with respect to the origin.

5. Graph Behavior:

   • For $x > 0$, $F(x) > 0$ (the function is positive).
   • For $x < 0$, $F(x) < 0$ (the function is negative).
   • The graph lies in the first and third quadrants.

Would you like a detailed graph or further exploration of this function?

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Related Questions:

1. What are some real-world applications of a function like $F(x) = \frac{2}{x}$?
2. How do vertical and horizontal asymptotes impact the graph of a rational function?
3. How does the graph of $\frac{2}{x}$ compare to $\frac{1}{x}$?
4. How would the domain and range change if $F(x) = \frac{2}{x+1}$?
5. What is the derivative of $F(x)$, and how does it relate to the slope of the graph?

Tip:

When working with rational functions, always analyze asymptotes and domain restrictions first, as they shape the overall graph and behavior of the function.