**Problem 3: Understanding Composite and Inverse Functions**Let $ S = \{1, 2, 3, 4\} $ and define the functions $ f, g : S \to S $ by $ f = \{(1, 3), (2, 2), (3, 4), (4, 1)\} $ and $ g = \{(1, 4), (2, 3), (3, 1), (4, 2)\} $. We aim to find the following compositions of functions:### (a) $ g^{-1} \circ f \circ g $### (b) $ f \circ g^{-1} \circ g $### (c) $ f^{-1} \circ g^{-1} \circ f \circ g $**Hints for Solving:**1. **Find Inverses $ g^{-1} $ and $ f^{-1} $:** - An inverse function $ g^{-1} $ reverses the mappings of $ g $, such that $ g^{-1}(b) = a $ if $ g(a) = b $. - Apply the same principle to find $ f^{-1} $.2. **Evaluate Composite Functions:** - Use the mappings $ f, g, g^{-1} $, and $ f^{-1} $ iteratively to find the required values for the compositions. - Ensure you follow the order of operations indicated by the compositions (from right to left).### Diagrams:No graphs or diagrams are provided in this problem statement. The mathematical relationships and operations are to be handled via the defined sets and compositions.Below are step-by-step breakdowns on how you might approach solving them.**For Detailed Solutions:**- Identify $ g^{-1} $ by reversing the pairs in $ g $.- Compose the functions as indicated in parts (a), (b), and (c).These tasks involve careful mapping and substitution to evaluate the final compositions. Explore decomposing and reconstituting functions systematically to derive the requested solutions.

Answered: Image /qna-images/answer/f732a7b4-3b03-49c3-9faf-25113dc9c498.jpg

Answered: 3. Let S = {1,2, 3, 4} and define…

3. Let S = {1,2, 3, 4} and define functions f, g : S → S by ƒ = {(1,3), (2,2), (3, 4), (4, 1)} and g %3D {(1,4), (2,3), (3, 1), (4, 2)}. Find (a) g¬l o f og (b) ƒ og¬1 og (c) f-1 o g¬1 o f og

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem 3: Understanding Composite and Inverse Functions**

Let $ S = \{1, 2, 3, 4\} $ and define the functions $ f, g : S \to S $ by $ f = \{(1, 3), (2, 2), (3, 4), (4, 1)\} $ and $ g = \{(1, 4), (2, 3), (3, 1), (4, 2)\} $.

We aim to find the following compositions of functions:

### (a) $ g^{-1} \circ f \circ g $

### (b) $ f \circ g^{-1} \circ g $

### (c) $ f^{-1} \circ g^{-1} \circ f \circ g $

**Hints for Solving:**
1. **Find Inverses $ g^{-1} $ and $ f^{-1} $:**
- An inverse function $ g^{-1} $ reverses the mappings of $ g $, such that $ g^{-1}(b) = a $ if $ g(a) = b $.
- Apply the same principle to find $ f^{-1} $.

2. **Evaluate Composite Functions:**
- Use the mappings $ f, g, g^{-1} $, and $ f^{-1} $ iteratively to find the required values for the compositions.
- Ensure you follow the order of operations indicated by the compositions (from right to left).

### Diagrams:
No graphs or diagrams are provided in this problem statement. The mathematical relationships and operations are to be handled via the defined sets and compositions.

Below are step-by-step breakdowns on how you might approach solving them.

**For Detailed Solutions:**

- Identify $ g^{-1} $ by reversing the pairs in $ g $.
- Compose the functions as indicated in parts (a), (b), and (c).

These tasks involve careful mapping and substitution to evaluate the final compositions. Explore decomposing and reconstituting functions systematically to derive the requested solutions.

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