Assignment: Find all intervals on which the function is positive and all intervals on which the function is negative: x-3 e²(x - 5)² √2x+1° f(x) Include a detailed explanation for each mathematical step written in grammatically correct complete sentences within a 2-column format.
Assignment: Find all intervals on which the function is positive and all intervals on which the function is negative: x-3 e²(x - 5)² √2x+1° f(x) Include a detailed explanation for each mathematical step written in grammatically correct complete sentences within a 2-column format.
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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solve please an example will be in the second photo
![### Determining the Zeros of \(f(x)\)
We start by determining all places within the domain of \(f(x)\) where \(f(x) = 0\). This occurs whenever the numerator of the fraction is zero. We solve this by setting each individual factor equal to zero and solving those equations.
#### Solving for Zeros:
1. \[(x + 3)\sqrt{4 - x} = 0\]
For \(x + 3 = 0\):
\[
x + 3 = 0 \implies x = -3
\]
For \(\sqrt{4 - x} = 0\):
\[
4 - x = 0 \implies x = 4
\]
Each solution, \(x = -3\) and \(x = 4\), lies within the domain of \(f(x)\). Therefore, \(f(x)\) has zeros at \(x = -3\) and \(x = 4\).
### Plotting Zeros on Number Line
The zero \(x = 4\) is already plotted on the number line since it is the boundary of the domain. The other zero, \(x = -3\), is added to the number line.
\[ \longleftarrow -3 \longrightarrow\quad 0 \quad \longrightarrow 4 \]
### Determining Sign in Each Interval
We choose test values in each interval to determine the sign of \(f(x)\). Test values are indicated in parentheses as they are not uniquely chosen.
#### Test Intervals and Values:
\[ (-\infty, -3) : \text{Test value} = -4 \]
\[ (-3, 0) : \text{Test value} = -1 \]
\[ (0, 4) : \text{Test value} = 1 \]
### Sign Analysis of Each Interval
1. **Interval \((-∞, -3)\):**
Test value \(x = -4\)
- Factor \( \sqrt{4 - x} \) is positive.
- Factor \( x^2 \) is positive.
- \( x + 3 \) is negative.
- \( x - 5 \) is negative.
Product: Positive (since two negatives make a positive).
2. **Interval \((-3, 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c60962f-c8d5-4255-8d11-ba7fb1b92564%2F97e58056-ef77-4fe4-a08f-acdedd005e2b%2Fa4s4rj9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Zeros of \(f(x)\)
We start by determining all places within the domain of \(f(x)\) where \(f(x) = 0\). This occurs whenever the numerator of the fraction is zero. We solve this by setting each individual factor equal to zero and solving those equations.
#### Solving for Zeros:
1. \[(x + 3)\sqrt{4 - x} = 0\]
For \(x + 3 = 0\):
\[
x + 3 = 0 \implies x = -3
\]
For \(\sqrt{4 - x} = 0\):
\[
4 - x = 0 \implies x = 4
\]
Each solution, \(x = -3\) and \(x = 4\), lies within the domain of \(f(x)\). Therefore, \(f(x)\) has zeros at \(x = -3\) and \(x = 4\).
### Plotting Zeros on Number Line
The zero \(x = 4\) is already plotted on the number line since it is the boundary of the domain. The other zero, \(x = -3\), is added to the number line.
\[ \longleftarrow -3 \longrightarrow\quad 0 \quad \longrightarrow 4 \]
### Determining Sign in Each Interval
We choose test values in each interval to determine the sign of \(f(x)\). Test values are indicated in parentheses as they are not uniquely chosen.
#### Test Intervals and Values:
\[ (-\infty, -3) : \text{Test value} = -4 \]
\[ (-3, 0) : \text{Test value} = -1 \]
\[ (0, 4) : \text{Test value} = 1 \]
### Sign Analysis of Each Interval
1. **Interval \((-∞, -3)\):**
Test value \(x = -4\)
- Factor \( \sqrt{4 - x} \) is positive.
- Factor \( x^2 \) is positive.
- \( x + 3 \) is negative.
- \( x - 5 \) is negative.
Product: Positive (since two negatives make a positive).
2. **Interval \((-3, 0
![---
### Show What You Know: Solving Inequalities with Various Function Types
**MAT 190 - Precalculus**
---
#### Objectives:
The purpose of this assignment is for you to:
1. Demonstrate your ability to solve inequalities with various function types.
2. Improve your mathematical writing to include full solutions with justifications.
3. Integrate mathematical statements into grammatically correct expositions.
---
#### Assignment:
Find all intervals on which the function is positive and all intervals on which the function is negative:
\[ f(x) = \frac{x - 3}{e^x (x - 5)^2 \sqrt{2x + 1}} \]
Include a detailed explanation for each mathematical step written in grammatically correct complete sentences within a 2-column format.
---
This transcription ensures clarity and precision for students tackling the assignment on solving inequalities with various function types in a structured manner.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c60962f-c8d5-4255-8d11-ba7fb1b92564%2F97e58056-ef77-4fe4-a08f-acdedd005e2b%2Fqod9zu3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Show What You Know: Solving Inequalities with Various Function Types
**MAT 190 - Precalculus**
---
#### Objectives:
The purpose of this assignment is for you to:
1. Demonstrate your ability to solve inequalities with various function types.
2. Improve your mathematical writing to include full solutions with justifications.
3. Integrate mathematical statements into grammatically correct expositions.
---
#### Assignment:
Find all intervals on which the function is positive and all intervals on which the function is negative:
\[ f(x) = \frac{x - 3}{e^x (x - 5)^2 \sqrt{2x + 1}} \]
Include a detailed explanation for each mathematical step written in grammatically correct complete sentences within a 2-column format.
---
This transcription ensures clarity and precision for students tackling the assignment on solving inequalities with various function types in a structured manner.
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