(a) \( y=\frac{-1}{x} \)

Certainly! Let's analyze the function provided in part (a): ### (a) \( y = \frac{-1}{x} \) #### 1. **Function Overview** The function \( y = \frac{-1}{x} \) is a rational function, specifically a type of hyperbola. It's a reciprocal function multiplied by -1, which affects its graph's orientation. #### 2. **Domain and Range** - **Domain**: All real numbers except \( x = 0 \) (since division by zero is undefined). \[ \text{Domain: } x \in \mathbb{R} \setminus \{0\} \] - **Range**: All real numbers except \( y = 0 \). \[ \text{Range: } y \in \mathbb{R} \setminus \{0\} \] #### 3. **Asymptotes** - **Vertical Asymptote**: \( x = 0 \) - The function approaches infinity or negative infinity as \( x \) approaches 0 from the right or left. - **Horizontal Asymptote**: \( y = 0 \) (the x-axis) - As \( x \) becomes very large (positively or negatively), \( y \) approaches 0. #### 4. **Graph Characteristics** - **Quadrants**: The graph of \( y = \frac{-1}{x} \) lies in the second and fourth quadrants because: - For \( x > 0 \), \( y = -\frac{1}{x} < 0 \) (4th quadrant). - For \( x < 0 \), \( y = -\frac{1}{x} > 0 \) (2nd quadrant). - **Symmetry**: - The graph is **symmetric** with respect to the **origin**. This means if you rotate the graph 180 degrees around the origin, it remains unchanged. Mathematically, this is known as **odd symmetry**, satisfying \( f(-x) = -f(x) \). #### 5. **Behavior Analysis** - **As \( x \to 0^+ \)**: \[ y = \frac{-1}{x} \to -\infty \] - **As \( x \to 0^- \)**: \[ y = \frac{-1}{x} \to +\infty \] - **As \( x \to +\infty \)**: \[ y = \frac{-1}{x} \to 0^- \] - **As \( x \to -\infty \)**: \[ y = \frac{-1}{x} \to 0^+ \] #### 6. **Plotting Points** | \( x \) | \( y = \frac{-1}{x} \) | |---------|------------------------| | -2 | 0.5 | | -1 | 1 | | -0.5 | 2 | | 0.5 | -2 | | 1 | -1 | | 2 | -0.5 | Plotting these points will help visualize the hyperbola's shape, confirming its placement in the second and fourth quadrants with the described asymptotes. #### 7. **Summary** The function \( y = \frac{-1}{x} \) is a hyperbola with: - **Vertical Asymptote** at \( x = 0 \) - **Horizontal Asymptote** at \( y = 0 \) - **Symmetry** about the origin - Located in the **second and fourth quadrants** Understanding these properties will aid in graphing the function and analyzing its behavior across different values of \( x \).