Let f and g be the functions in the table below. g(x) 2 3 1 X 1 2 3 f(x) 3 1 2 (a) If F(x) = f(f(x)), find F (3). F '(3) = (b) If G(x) = g(g(x)), find G'(2). G'(2) = f'(x) 4 5 7 st g'(x) 6 7 9
Let f and g be the functions in the table below. g(x) 2 3 1 X 1 2 3 f(x) 3 1 2 (a) If F(x) = f(f(x)), find F (3). F '(3) = (b) If G(x) = g(g(x)), find G'(2). G'(2) = f'(x) 4 5 7 st g'(x) 6 7 9
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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![**Let \( f \) and \( g \) be the functions in the table below.**
| \( x \) | \( f(x) \) | \( g(x) \) | \( f'(x) \) | \( g'(x) \) |
|:------:|:-------:|:-------:|:-------:|:-------:|
| 1 | 3 | 2 | 4 | 6 |
| 2 | 1 | 3 | 5 | 7 |
| 3 | 2 | 1 | 7 | 9 |
### Problems
**(a)** If \( F(x) = f(f(x)) \), find \( F'(3) \).
\[ F'(3) = \_\_\_\_\_\_\_ \]
**(b)** If \( G(x) = g(g(x)) \), find \( G'(2) \).
\[ G'(2) = \_\_\_\_\_\_\_ \]
### Solution Steps
**For (a):**
1. Identify the function composition \( F(x) = f(f(x)) \).
2. Use the chain rule to differentiate \( F(x) \). That is, \( F'(x) = f'(f(x)) \cdot f'(x) \).
3. Determine \( f(3) \) from the table. From the table, \( f(3) = 2 \).
4. Find \( f'(2) \) and \( f'(3) \) from the table.
- \( f'(2) = 5 \)
- \( f'(3) = 7 \)
5. Calculate \( F'(3) = f'(f(3)) \cdot f'(3) = f'(2) \cdot f'(3) = 5 \cdot 7 = 35 \).
**So,**
\[ F'(3) = 35 \]
**For (b):**
1. Identify the function composition \( G(x) = g(g(x)) \).
2. Use the chain rule to differentiate \( G(x) \). That is, \( G'(x) = g'(g(x)) \cdot g'(x) \).
3. Determine \( g(2) \) from the table. From the table,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80fd0a7-9ae8-4bc5-b926-d69bfc10c0d7%2F538c1972-987f-40e8-ad97-c927849bb51c%2Fnanur8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Let \( f \) and \( g \) be the functions in the table below.**
| \( x \) | \( f(x) \) | \( g(x) \) | \( f'(x) \) | \( g'(x) \) |
|:------:|:-------:|:-------:|:-------:|:-------:|
| 1 | 3 | 2 | 4 | 6 |
| 2 | 1 | 3 | 5 | 7 |
| 3 | 2 | 1 | 7 | 9 |
### Problems
**(a)** If \( F(x) = f(f(x)) \), find \( F'(3) \).
\[ F'(3) = \_\_\_\_\_\_\_ \]
**(b)** If \( G(x) = g(g(x)) \), find \( G'(2) \).
\[ G'(2) = \_\_\_\_\_\_\_ \]
### Solution Steps
**For (a):**
1. Identify the function composition \( F(x) = f(f(x)) \).
2. Use the chain rule to differentiate \( F(x) \). That is, \( F'(x) = f'(f(x)) \cdot f'(x) \).
3. Determine \( f(3) \) from the table. From the table, \( f(3) = 2 \).
4. Find \( f'(2) \) and \( f'(3) \) from the table.
- \( f'(2) = 5 \)
- \( f'(3) = 7 \)
5. Calculate \( F'(3) = f'(f(3)) \cdot f'(3) = f'(2) \cdot f'(3) = 5 \cdot 7 = 35 \).
**So,**
\[ F'(3) = 35 \]
**For (b):**
1. Identify the function composition \( G(x) = g(g(x)) \).
2. Use the chain rule to differentiate \( G(x) \). That is, \( G'(x) = g'(g(x)) \cdot g'(x) \).
3. Determine \( g(2) \) from the table. From the table,
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