The equation $y = \frac{3}{x}$ represents a hyperbola, which is a type of rational function.

Key Points:

1. Domain: The function is undefined when $x = 0$ because division by zero is undefined. Therefore, the domain is all real numbers except $x = 0$.
2. Range: The range of the function is all real numbers except $y = 0$ because the function never touches or crosses the x-axis.
3. Asymptotes:
   • Vertical asymptote: $x = 0$.
   • Horizontal asymptote: $y = 0$.
4. Behavior: As $x$ approaches zero from the positive side, $y$ increases without bound (goes to infinity). As $x$ approaches zero from the negative side, $y$ decreases without bound (goes to negative infinity).
5. Symmetry: The graph is symmetric with respect to the origin, meaning it is unchanged if you rotate it 180 degrees around the origin.

Would you like more details or have any questions?

Related Questions:

1. How does the graph of $y = \frac{3}{x}$ compare to $y = \frac{1}{x}$?
2. What happens to the function if we add a constant, such as $y = \frac{3}{x} + 2$?
3. Can you explain the concept of vertical and horizontal asymptotes in more detail?
4. How would you solve the equation $\frac{3}{x} = 2$ for $x$?
5. What is the significance of the symmetry in the graph of $y = \frac{3}{x}$?

Tip:

When analyzing the behavior of a rational function, always check for vertical and horizontal asymptotes to understand how the function behaves as $x$ becomes very large or very small.