**Identifying Intervals of Increase and Decrease for Rational Functions**In this lesson, we will analyze the graph of a rational function to determine the intervals over which the function is increasing and decreasing. Below is a detailed explanation of the graph provided.### Graph Description:The graph presented is of a rational function. It shows two distinct branches which are not connected, indicating asymptotic behavior. The function appears to have a vertical asymptote at \( x = 0 \) and horizontal asymptotes as \( y \) approaches 0 when \( x \) tends to infinity or negative infinity.### Characteristics of the Graph:1. **Asymptotic Behavior**: - **Vertical Asymptote**: The graph approaches but never touches or crosses the line \( x = 0 \). This indicates a vertical asymptote at \( x = 0 \). - **Horizontal Asymptote**: The graph approaches \( y = 0 \) as \( x \) tends towards \( \infty \) or \( -\infty \). Therefore, the horizontal asymptote is at \( y = 0 \).2. **Intervals of Increase**: - For \( x < 0 \): The function decreases as \( x \) approaches 0 from the left, which suggests the function is not increasing in this interval. - For \( x > 0 \): Similarly, as \( x \) moves away from 0 to the right, the function is also decreasing and thus not increasing in this interval as well.3. **Intervals of Decrease**: - For \( x < 0 \): As \( x \) approaches 0 from the left, the graph goes downwards, showing the function is decreasing in \( (-\infty, 0) \). - For \( x > 0 \): As \( x \) moves away from 0 towards the right, the graph goes down, signifying the function is also decreasing in \( (0, \infty) \).The rational function is consistently decreasing over all regions except precisely at \( x = 0 \), where it is undefined due to the vertical asymptote.### Key Takeaways- The function decreases on \( (-\infty, 0) \) and \( (0, \infty) \).- The function increases on no interval.- It has a vertical asym

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Identify the intervals of increase and decrease for the given rational function graph. 0

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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**Identifying Intervals of Increase and Decrease for Rational Functions**

In this lesson, we will analyze the graph of a rational function to determine the intervals over which the function is increasing and decreasing. Below is a detailed explanation of the graph provided.

### Graph Description:

The graph presented is of a rational function. It shows two distinct branches which are not connected, indicating asymptotic behavior. The function appears to have a vertical asymptote at \( x = 0 \) and horizontal asymptotes as \( y \) approaches 0 when \( x \) tends to infinity or negative infinity.

### Characteristics of the Graph:

1. **Asymptotic Behavior**:
- **Vertical Asymptote**: The graph approaches but never touches or crosses the line \( x = 0 \). This indicates a vertical asymptote at \( x = 0 \).
- **Horizontal Asymptote**: The graph approaches \( y = 0 \) as \( x \) tends towards \( \infty \) or \( -\infty \). Therefore, the horizontal asymptote is at \( y = 0 \).

2. **Intervals of Increase**:
- For \( x < 0 \): The function decreases as \( x \) approaches 0 from the left, which suggests the function is not increasing in this interval.
- For \( x > 0 \): Similarly, as \( x \) moves away from 0 to the right, the function is also decreasing and thus not increasing in this interval as well.

3. **Intervals of Decrease**:
- For \( x < 0 \): As \( x \) approaches 0 from the left, the graph goes downwards, showing the function is decreasing in \( (-\infty, 0) \).
- For \( x > 0 \): As \( x \) moves away from 0 towards the right, the graph goes down, signifying the function is also decreasing in \( (0, \infty) \).

The rational function is consistently decreasing over all regions except precisely at \( x = 0 \), where it is undefined due to the vertical asymptote.

### Key Takeaways
- The function decreases on \( (-\infty, 0) \) and \( (0, \infty) \).
- The function increases on no interval.
- It has a vertical asym

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