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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![**Transcription and Explanation for Educational Website**
---
**Text Transcription:**
"The graphs of \( f \) and \( g \) are given. Find a formula for the function \( g \).
\[ g(x) = \underline{\hspace{3cm}} \]"
**Graph Analysis:**
The graph depicts two functions, \( f(x) = |x| \) and \( g(x) \).
- **Function \( f(x) = |x| \)**:
- This is represented by the blue V-shaped graph.
- It has a vertex at the origin (0, 0).
- For \( x \geq 0 \), \( f(x) = x \).
- For \( x < 0 \), \( f(x) = -x \).
- **Function \( g(x) \)**:
- This is represented by the red V-shaped graph.
- It is vertically stretched compared to \( f(x) = |x| \).
- The slopes of the lines passing through the vertex are steeper than those of \( f(x) \).
**Understanding \( g(x) \):**
To derive \( g(x) \), observe that the graph of \( g \) has the same V-shape as \( f(x) = |x| \) but is stretched vertically. The magnitude of this stretch can be determined by comparing slopes:
- The slope of \( f(x) = |x| \) is 1 in both the positive and negative directions.
- The slope of \( g(x) \) is observed to be 2 in the positive and negative directions, which indicates that \( g(x) \) is a vertical stretch by a factor of 2 of \( f(x) \).
**Conclusion:**
\[ g(x) = 2|x| \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e588b9-b272-406a-8435-8ef7027d16be%2Fdc7a7265-5993-4e76-846b-780c121e3c81%2F8nncgs_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation for Educational Website**
---
**Text Transcription:**
"The graphs of \( f \) and \( g \) are given. Find a formula for the function \( g \).
\[ g(x) = \underline{\hspace{3cm}} \]"
**Graph Analysis:**
The graph depicts two functions, \( f(x) = |x| \) and \( g(x) \).
- **Function \( f(x) = |x| \)**:
- This is represented by the blue V-shaped graph.
- It has a vertex at the origin (0, 0).
- For \( x \geq 0 \), \( f(x) = x \).
- For \( x < 0 \), \( f(x) = -x \).
- **Function \( g(x) \)**:
- This is represented by the red V-shaped graph.
- It is vertically stretched compared to \( f(x) = |x| \).
- The slopes of the lines passing through the vertex are steeper than those of \( f(x) \).
**Understanding \( g(x) \):**
To derive \( g(x) \), observe that the graph of \( g \) has the same V-shape as \( f(x) = |x| \) but is stretched vertically. The magnitude of this stretch can be determined by comparing slopes:
- The slope of \( f(x) = |x| \) is 1 in both the positive and negative directions.
- The slope of \( g(x) \) is observed to be 2 in the positive and negative directions, which indicates that \( g(x) \) is a vertical stretch by a factor of 2 of \( f(x) \).
**Conclusion:**
\[ g(x) = 2|x| \]
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