**Mathematics: Analyzing Functions****Objective:**Understand how to determine the open intervals where a given function is increasing, decreasing, or constant.**Problem:**Find the open interval(s) where the following function is increasing, decreasing, or constant. Express your answer in interval notation.\[ r(x) = -\frac{1}{x-4} \]**Instructions:**1. Analyze the given function $ r(x) = -\frac{1}{x-4} $.2. Determine the critical points by finding the derivative $ r'(x) $.3. Identify where the function is increasing, decreasing, or constant.4. Express the intervals in interval notation.**Approach:**1. **Find the Derivative:** Calculate the first derivative of $ r(x) $ to understand the behavior of the function. \[ r'(x) = \frac{d}{dx} \left( -\frac{1}{x-4} \right) \]2. **Determine Critical Points:** Identify values of $ x $ where $ r'(x) = 0 $ or $ r'(x) $ does not exist.3. **Analyze Intervals:** Study the sign of $ r'(x) $ in each interval to determine whether the function is increasing, decreasing, or constant.4. **Express in Interval Notation:** Provide the intervals in the form of (a, b), where applicable.*Note:* The function $ r(x) = -\frac{1}{x-4} $ has a vertical asymptote at $ x = 4 $, and the domain of the function is $ x \neq 4 $.**Diagram/Graph:**If a graph were provided, it would show a hyperbola with a vertical asymptote at $ x = 4 $. The left branch (for $ x < 4 $) would approach $ x = 4 $ from below and $ y = 0 $ from above. The right branch (for $ x > 4 $) would approach $ x = 4 $ from above and $ y = 0 $ from below.**Solution:**Through calculation and analysis:- $ r(x) $ is decreasing for $ x < 4 $.- $ r(x) $ is also ### Interval Behavior OptionsSelecting an option will display any text boxes needed to complete your answer:- ⃝ Increasing on one interval- ⃝ Decreasing on one interval- ⃝ Constant on one interval- ⃝ Increasing on one interval and Decreasing on another- ⃝ Increasing on two intervals- ⃝ Decreasing on two intervalsThese options allow you to specify the behavior of a function or a variable over different intervals. Here’s a brief explanation of each option:1. **Increasing on one interval**: The function or variable increases (goes up) over a specific interval.2. **Decreasing on one interval**: The function or variable decreases (goes down) over a specific interval.3. **Constant on one interval**: The function or variable remains constant (unchanged) over a specific interval.4. **Increasing on one interval and Decreasing on another**: The function or variable increases over one interval and decreases over another interval.5. **Increasing on two intervals**: The function or variable increases over two separate intervals.6. **Decreasing on two intervals**: The function or variable decreases over two separate intervals.

Answered: Image /qna-images/answer/9c979e01-448b-494f-88b1-e13b1a807348.jpg

Answered: -= Find the open interval(s) where the…

-= Find the open interval(s) where the following function is increasing, decreasing, or constant. Express your answer in interval notation.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Question

**Mathematics: Analyzing Functions**

**Objective:**
Understand how to determine the open intervals where a given function is increasing, decreasing, or constant.

**Problem:**
Find the open interval(s) where the following function is increasing, decreasing, or constant. Express your answer in interval notation.

\[ r(x) = -\frac{1}{x-4} \]

**Instructions:**
1. Analyze the given function $ r(x) = -\frac{1}{x-4} $.
2. Determine the critical points by finding the derivative $ r'(x) $.
3. Identify where the function is increasing, decreasing, or constant.
4. Express the intervals in interval notation.

**Approach:**
1. **Find the Derivative:**
Calculate the first derivative of $ r(x) $ to understand the behavior of the function.

\[ r'(x) = \frac{d}{dx} \left( -\frac{1}{x-4} \right) \]

2. **Determine Critical Points:**
Identify values of $ x $ where $ r'(x) = 0 $ or $ r'(x) $ does not exist.

3. **Analyze Intervals:**
Study the sign of $ r'(x) $ in each interval to determine whether the function is increasing, decreasing, or constant.

4. **Express in Interval Notation:**
Provide the intervals in the form of (a, b), where applicable.

*Note:* The function $ r(x) = -\frac{1}{x-4} $ has a vertical asymptote at $ x = 4 $, and the domain of the function is $ x \neq 4 $.

**Diagram/Graph:**
If a graph were provided, it would show a hyperbola with a vertical asymptote at $ x = 4 $. The left branch (for $ x < 4 $) would approach $ x = 4 $ from below and $ y = 0 $ from above. The right branch (for $ x > 4 $) would approach $ x = 4 $ from above and $ y = 0 $ from below.

**Solution:**
Through calculation and analysis:

- $ r(x) $ is decreasing for $ x < 4 $.
- $ r(x) $ is also

### Interval Behavior Options

Selecting an option will display any text boxes needed to complete your answer:

- ⃝ Increasing on one interval
- ⃝ Decreasing on one interval
- ⃝ Constant on one interval
- ⃝ Increasing on one interval and Decreasing on another
- ⃝ Increasing on two intervals
- ⃝ Decreasing on two intervals

These options allow you to specify the behavior of a function or a variable over different intervals. Here’s a brief explanation of each option:

1. **Increasing on one interval**: The function or variable increases (goes up) over a specific interval.

2. **Decreasing on one interval**: The function or variable decreases (goes down) over a specific interval.

3. **Constant on one interval**: The function or variable remains constant (unchanged) over a specific interval.

4. **Increasing on one interval and Decreasing on another**: The function or variable increases over one interval and decreases over another interval.

5. **Increasing on two intervals**: The function or variable increases over two separate intervals.

6. **Decreasing on two intervals**: The function or variable decreases over two separate intervals.

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