Use the graph of f(x) = ² below to find a number & such that 2-1 < 0.2 whenever 0 < |x-1|< 8. 1.2 y 1 0.8- Round your answer to four decimal places. 8 = 1
Use the graph of f(x) = ² below to find a number & such that 2-1 < 0.2 whenever 0 < |x-1|< 8. 1.2 y 1 0.8- Round your answer to four decimal places. 8 = 1
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter1: Expressions And Functions
Section: Chapter Questions
Problem 74SGR
Related questions
Question
![## Definition and Illustration of Continuous Functions
### Using Graph Analysis for a Given Function
We use the graph of \( f(x) = x^2 \) below to find a number \( \delta \) such that
\[
|x^2 - 1| < 0.2 \text{ whenever } 0 < |x - 1| < \delta.
\]
#### Diagram Explanation
The graph provided is a plot of the function \( f(x) = x^2 \). On this graph:
- The horizontal axis represents the \( x \)-values, and the vertical axis represents the \( y \)-values.
- The blue curve represents the function \( f(x) = x^2 \).
- A black horizontal line intersects the curve at \( (1, 1) \), indicating the point where \( y = 1 \).
- Red horizontal lines are drawn at \( y = 1.2 \) and \( y = 0.8 \). These lines show the range within which \( |x^2 - 1| < 0.2 \).
From the graph:
- The vertical distance between \( f(x) \) and \( y = 1 \) must be less than 0.2 for these bounds.
#### Finding \( \delta \)
- Vertical black lines mark the corresponding \( x \)-values at which the horizontal lines \( y = 1.2 \) and \( y = 0.8 \) intersect the graph of \( f(x) \).
- The \( x \)-coordinates of these intersections determine the values within which \( x \) must lie.
For the specific conditions given:
\[
\delta = |x - 1|
\]
Find \( x \) such that \( 0 < |x - 1| < \delta \).
Finally, the problem states:
"Round your answer to four decimal places."
\[
\delta = \_\_\_\_\_\_\_\_\_
\]
Insert the appropriate value of \( \delta \) from the above graphical analysis.
---
The graphical method outlined helps visualize the concept of continuity and provides a means to find precise \(\delta\)-values based on \( \epsilon \)-arguments in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4b14bdb-fbdf-4a4d-b0e2-53699d6b8365%2F7f7f6ba7-ad96-4e4f-a995-e772c8dc6978%2F1qstf5l_processed.png&w=3840&q=75)
Transcribed Image Text:## Definition and Illustration of Continuous Functions
### Using Graph Analysis for a Given Function
We use the graph of \( f(x) = x^2 \) below to find a number \( \delta \) such that
\[
|x^2 - 1| < 0.2 \text{ whenever } 0 < |x - 1| < \delta.
\]
#### Diagram Explanation
The graph provided is a plot of the function \( f(x) = x^2 \). On this graph:
- The horizontal axis represents the \( x \)-values, and the vertical axis represents the \( y \)-values.
- The blue curve represents the function \( f(x) = x^2 \).
- A black horizontal line intersects the curve at \( (1, 1) \), indicating the point where \( y = 1 \).
- Red horizontal lines are drawn at \( y = 1.2 \) and \( y = 0.8 \). These lines show the range within which \( |x^2 - 1| < 0.2 \).
From the graph:
- The vertical distance between \( f(x) \) and \( y = 1 \) must be less than 0.2 for these bounds.
#### Finding \( \delta \)
- Vertical black lines mark the corresponding \( x \)-values at which the horizontal lines \( y = 1.2 \) and \( y = 0.8 \) intersect the graph of \( f(x) \).
- The \( x \)-coordinates of these intersections determine the values within which \( x \) must lie.
For the specific conditions given:
\[
\delta = |x - 1|
\]
Find \( x \) such that \( 0 < |x - 1| < \delta \).
Finally, the problem states:
"Round your answer to four decimal places."
\[
\delta = \_\_\_\_\_\_\_\_\_
\]
Insert the appropriate value of \( \delta \) from the above graphical analysis.
---
The graphical method outlined helps visualize the concept of continuity and provides a means to find precise \(\delta\)-values based on \( \epsilon \)-arguments in calculus.
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