The equation \( y = \frac{1}{x} \) represents a **hyperbola** in the coordinate plane, with the following characteristics:

1. **Asymptotes**: 
   - The function approaches the x-axis (\( y = 0 \)) as \( x \to \infty \) or \( x \to -\infty \).
   - The function approaches the y-axis (\( x = 0 \)) but never touches it, creating a vertical asymptote at \( x = 0 \).

2. **Domain**: 
   - The function is defined for all real numbers except \( x = 0 \), i.e., \( x \in (-\infty, 0) \cup (0, \infty) \).

3. **Range**: 
   - The range is also all real numbers except \( y = 0 \), i.e., \( y \in (-\infty, 0) \cup (0, \infty) \).

4. **Symmetry**: 
   - The graph is symmetric about the origin, meaning it's an **odd function**: \( f(-x) = -f(x) \).

5. **Behavior**: 
   - For positive \( x \), as \( x \) increases, \( y \) decreases.
   - For negative \( x \), as \( x \) decreases, \( y \) also decreases in magnitude but stays negative.

Would you like more details or any specific questions about this function?

### Related questions:
1. What are the vertical and horizontal asymptotes of a function?
2. How do you determine the domain of rational functions?
3. Can you explain how to find symmetry in functions?
4. What are the transformations of \( y = \frac{1}{x} \)?
5. How do you sketch the graph of \( y = \frac{1}{x} \)?

**Tip**: Always remember that rational functions like \( y = \frac{1}{x} \) can have vertical asymptotes where the denominator is zero!

Explore the properties of the rational function y = 1/x, including its domain, range, asymptotes, and symmetry. This problem is ideal for students in Grades 9-12 looking to deepen their understanding of algebraic functions and graph behavior.

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