Find the inverse of f. Then use a graphing utility to plot the graphs off and f-¹ on the same set of axes. f(x) = 1- & 1 + e
Find the inverse of f. Then use a graphing utility to plot the graphs off and f-¹ on the same set of axes. f(x) = 1- & 1 + e
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I need help with this problem and an explanation of this problem.
![**Topic: Finding and Plotting the Inverse of a Function**
**Objective:**
Learn how to find the inverse of the function \( f \). Use a graphing utility to plot both the function \( f \) and its inverse \( f^{-1} \) on the same set of axes.
**Given Function:**
\[ f(x) = \frac{1 - e^{8x}}{1 + e^{8x}} \]
### Steps to Find the Inverse:
1. **Rewrite the function equation by replacing \( f(x) \) with \( y \).**
\[ y = \frac{1 - e^{8x}}{1 + e^{8x}} \]
2. **Interchange \( x \) and \( y \).**
\[ x = \frac{1 - e^{8y}}{1 + e^{8y}} \]
3. **Solve for \( y \) to find the inverse function.**
4. **Express the inverse function in terms of \( x \), denoted as \( f^{-1}(x) \).**
### Using a Graphing Utility:
1. **Plot the original function \( f(x) \).**
2. **Plot the inverse function \( f^{-1}(x) \).**
3. **Ensure both graphs are on the same set of axes for comparison.**
### Graph Explanation:
- **Axes:**
- The horizontal axis (x-axis) represents the input variable \( x \).
- The vertical axis (y-axis) represents the output variable \( y \) or \( f(x) \).
- **Graph of \( f(x) \):**
- Observe how the function behaves. For the given function \( f(x) \), note the key points and asymptotic behavior.
- **Graph of \( f^{-1}(x) \):**
- The inverse function will mirror the original function across the line \( y = x \).
- Plot \( f^{-1}(x) \) to observe how it relates to \( f(x) \).
### Learning Outcome:
**Understand the relationship between a function and its inverse by graphically plotting both on the same axes. This exercise helps visualize how the inverse function is a reflection of the original function over the line \( y = x \).**
**Note:**
Graphical tools like Desmos, GeoGebra, or graphing calculators can be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e135d5c-f867-4afd-9f0d-b505f4f19664%2F0d13e6ed-0423-44a8-9679-de1bd5cf9cc0%2F421yp7_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Finding and Plotting the Inverse of a Function**
**Objective:**
Learn how to find the inverse of the function \( f \). Use a graphing utility to plot both the function \( f \) and its inverse \( f^{-1} \) on the same set of axes.
**Given Function:**
\[ f(x) = \frac{1 - e^{8x}}{1 + e^{8x}} \]
### Steps to Find the Inverse:
1. **Rewrite the function equation by replacing \( f(x) \) with \( y \).**
\[ y = \frac{1 - e^{8x}}{1 + e^{8x}} \]
2. **Interchange \( x \) and \( y \).**
\[ x = \frac{1 - e^{8y}}{1 + e^{8y}} \]
3. **Solve for \( y \) to find the inverse function.**
4. **Express the inverse function in terms of \( x \), denoted as \( f^{-1}(x) \).**
### Using a Graphing Utility:
1. **Plot the original function \( f(x) \).**
2. **Plot the inverse function \( f^{-1}(x) \).**
3. **Ensure both graphs are on the same set of axes for comparison.**
### Graph Explanation:
- **Axes:**
- The horizontal axis (x-axis) represents the input variable \( x \).
- The vertical axis (y-axis) represents the output variable \( y \) or \( f(x) \).
- **Graph of \( f(x) \):**
- Observe how the function behaves. For the given function \( f(x) \), note the key points and asymptotic behavior.
- **Graph of \( f^{-1}(x) \):**
- The inverse function will mirror the original function across the line \( y = x \).
- Plot \( f^{-1}(x) \) to observe how it relates to \( f(x) \).
### Learning Outcome:
**Understand the relationship between a function and its inverse by graphically plotting both on the same axes. This exercise helps visualize how the inverse function is a reflection of the original function over the line \( y = x \).**
**Note:**
Graphical tools like Desmos, GeoGebra, or graphing calculators can be
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 13 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
