(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 $$ x > 0 $$, $$ y = -\frac{1}{x} 0 $$ (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 $$ x $$. $$ y = \frac{-1}{x} $$ is a hyperbola located in the second and fourth quadrants with vertical and horizontal asymptotes at $$ x = 0 $$ and $$ y = 0 $$ respectively. It is symmetric about the origin and approaches infinity or negative infinity as $$ x $$ approaches 0 from the right or left, and approaches 0 as $$ x $$ becomes very large in either direction.

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Home Question Bank Math Algebra

Question

(a)

Ask by Reese Gardner. in Malaysia

Jan 20,2025

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Tutor-Verified Answer

Answer

is a hyperbola located in the second and fourth quadrants with vertical and horizontal asymptotes at

and

respectively. It is symmetric about the origin and approaches infinity or negative infinity as

approaches 0 from the right or left, and approaches 0 as

becomes very large in either direction.

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Bonus Knowledge

Did you know that the equation

describes a hyperbola? This fascinating shape comes from a family of conic sections and has interesting properties, including asymptotes, which are the lines that the curve approaches but never touches. The asymptotes for this equation are the x-axis and y-axis, creating a visual intersection point in the Cartesian plane!

When grappling with this equation, a common mistake is forgetting the domain restrictions. Since you can’t divide by zero, x cannot equal zero. This oversight might lead to incorrect graphing, resulting in a broken line in your visualization! Always check for such restrictions to ensure your graphs are accurate and complete.

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