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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Question
Consider the following functions.
Sketch the region bounded by the graphs of the functions.
![### Analyzing Bounded Regions by Functions
#### Problem Statement:
Consider the following functions:
\[ f(x) = \sqrt[3]{x - 7} \]
\[ g(x) = x - 7 \]
#### Objective:
Sketch the region bounded by the graphs of the functions.
#### Graphical Analysis:
The image presents four graphs, each plotting the region bounded by the given functions \( f(x) \) and \( g(x) \). Here's a detailed description of each graph:
**Top-Left Graph:**
- The \(x\)-axis ranges from 0 to 8.
- The \(y\)-axis ranges from -2 to 2.
- The graph of \( f(x) \) is a cubic root function shifted right by 7 units.
- The graph of \( g(x) \) is a linear function also shifted downwards.
- The bounded region is shaded in blue between the curves.
**Top-Right Graph:**
- The \(x\)-axis ranges from -8 to 0.
- The \(y\)-axis ranges from -2 to 2.
- The graph of \( f(x) \) again shows a cubic root function, but reflected.
- The graph of \( g(x) \) intersects similarly as in the top-left graph.
- The bounded region is shaded in blue.
**Bottom-Left Graph:**
- The \(x\)-axis ranges from 0 to 8.
- The \(y\)-axis ranges from -2 to 2.
- The graphs of \( f(x) \) and \( g(x) \) appear similar to the top-left graph but are extending differently.
- The shaded area is still between the two curves.
**Bottom-Right Graph:**
- The \(x\)-axis ranges from -8 to 0.
- The \(y\)-axis ranges from -2 to 2.
- The graph of \( f(x) \) mirrors the bottom-left graph.
- The graph of \( g(x) \) follows the same pattern.
- The blue region represents the bounded area.
#### Task:
After analyzing the graphs, calculate the area of the region bounded by \( f(x) \) and \( g(x) \).
**Find the area of the region.**
\[ \boxed{\phantom{0}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda6dd950-15a2-4e47-84e9-a8f156faff1f%2F38dca044-cbcd-4ccd-b5d8-11a90b364ee5%2F6p0bsxc_processed.png&w=3840&q=75)
Transcribed Image Text:### Analyzing Bounded Regions by Functions
#### Problem Statement:
Consider the following functions:
\[ f(x) = \sqrt[3]{x - 7} \]
\[ g(x) = x - 7 \]
#### Objective:
Sketch the region bounded by the graphs of the functions.
#### Graphical Analysis:
The image presents four graphs, each plotting the region bounded by the given functions \( f(x) \) and \( g(x) \). Here's a detailed description of each graph:
**Top-Left Graph:**
- The \(x\)-axis ranges from 0 to 8.
- The \(y\)-axis ranges from -2 to 2.
- The graph of \( f(x) \) is a cubic root function shifted right by 7 units.
- The graph of \( g(x) \) is a linear function also shifted downwards.
- The bounded region is shaded in blue between the curves.
**Top-Right Graph:**
- The \(x\)-axis ranges from -8 to 0.
- The \(y\)-axis ranges from -2 to 2.
- The graph of \( f(x) \) again shows a cubic root function, but reflected.
- The graph of \( g(x) \) intersects similarly as in the top-left graph.
- The bounded region is shaded in blue.
**Bottom-Left Graph:**
- The \(x\)-axis ranges from 0 to 8.
- The \(y\)-axis ranges from -2 to 2.
- The graphs of \( f(x) \) and \( g(x) \) appear similar to the top-left graph but are extending differently.
- The shaded area is still between the two curves.
**Bottom-Right Graph:**
- The \(x\)-axis ranges from -8 to 0.
- The \(y\)-axis ranges from -2 to 2.
- The graph of \( f(x) \) mirrors the bottom-left graph.
- The graph of \( g(x) \) follows the same pattern.
- The blue region represents the bounded area.
#### Task:
After analyzing the graphs, calculate the area of the region bounded by \( f(x) \) and \( g(x) \).
**Find the area of the region.**
\[ \boxed{\phantom{0}} \]
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