The relationship `y = (3)/(x)` is best represent at :

Options

A.
B.
C.
D.

Answer

To solve the problem of representing the relationship \( y = \frac{3}{x} \), we will analyze the behavior of the function and plot its graph step by step.

### Step 1: Understand the function
The given function is \( y = \frac{3}{x} \). This is a hyperbolic function, which means it will have certain characteristics based on the values of \( x \).

**Hint:** Consider the behavior of the function as \( x \) approaches different values, especially zero and infinity.

### Step 2: Analyze the function at critical points
1. **When \( x = 0 \)**:
- The function \( y = \frac{3}{x} \) is undefined because division by zero is not possible. Therefore, \( y \) approaches infinity as \( x \) approaches zero from the right (positive side).
- **Conclusion**: As \( x \to 0^+ \), \( y \to +\infty \).

2. **When \( x \) is positive and approaches infinity**:
- As \( x \) increases, \( y = \frac{3}{x} \) approaches zero. This means that for very large values of \( x \), \( y \) becomes very small.
- **Conclusion**: As \( x \to +\infty \), \( y \to 0 \).

3. **When \( x \) is negative**:
- If \( x \) is negative, \( y \) will also be negative. For example, if \( x = -1 \), then \( y = -3 \); if \( x = -2 \), then \( y = -1.5 \), and so on.
- As \( x \) approaches zero from the left (negative side), \( y \) approaches negative infinity.
- **Conclusion**: As \( x \to 0^- \), \( y \to -\infty \).

### Step 3: Graph the function
Now that we have analyzed the behavior of the function, we can plot it:
- The graph will have two branches: one in the first quadrant (where \( x > 0 \) and \( y > 0 \)) and one in the third quadrant (where \( x < 0 \) and \( y < 0 \)).
- The graph will approach the x-axis (y = 0) as \( x \) goes to infinity and will approach the y-axis (x = 0) as \( y \) goes to infinity.

### Step 4: Identify the correct representation
From the analysis, we can conclude that:
- The function has vertical asymptote at \( x = 0 \) (where \( y \) approaches infinity).
- The function has horizontal asymptote at \( y = 0 \) (as \( x \) approaches infinity).

Therefore, the best representation of the relationship \( y = \frac{3}{x} \) is a hyperbola that opens towards the first and third quadrants.

**Final Conclusion**: The correct option that represents the relationship \( y = \frac{3}{x} \) is the one that shows the hyperbolic nature with the asymptotes as described.