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Use the given graph of f(x) = √x to find a number & such that if |x - 4| < & then √x - 2 <0.4. 8 = y 2.4 2 1.6 ? 4 y = √x ?

Use the given graph of f(x) = √x to find a number & such that if |x - 4| < & then √x - 2 <0.4. 8 = y 2.4 2 1.6 ? 4 y = √x ?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 58E
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Question

Q6. Please answer this question

The image presents a mathematical problem involving the function \( f(x) = \sqrt{x} \). It asks to find a number \( \delta \) such that if \( |x - 4| < \delta \), then \( |\sqrt{x} - 2| < 0.4 \). ### Graph Explanation: - **Axes**: The graph is plotted on the Cartesian plane with the x-axis labeled as \( x \) and the y-axis labeled as \( y \). - **Function Line**: The curve represents the function \( y = \sqrt{x} \), which steadily increases as \( x \) increases. - **Horizontal Lines**: Two horizontal green lines are drawn at \( y = 2.4 \) and \( y = 1.6 \). These lines are used to represent the bounds within \( |\sqrt{x} - 2| < 0.4 \). - **Vertical Lines**: Two vertical lines intersect the function curve at the points corresponding to \( y = 2.4 \) and \( y = 1.6 \). The x-coordinates of these intersections suggest the values for \( ? \) in question, and you are asked to identify these values related to \( x = 4 \). - **Target Point**: The point on the curve where \( x = 4 \) corresponds to \( \sqrt{4} = 2 \). ### Task: You need to define the distance \( \delta \) around \( x = 4 \) such that all \( x \) within this range (\( |x - 4| < \delta \)) have corresponding \( y \)-values on the graph within the range \( 1.6 < y < 2.4 \). ### Solution Box: - A box labeled \(\delta =\) is provided to input the calculated \( \delta \).
Transcribed Image Text:The image presents a mathematical problem involving the function \( f(x) = \sqrt{x} \). It asks to find a number \( \delta \) such that if \( |x - 4| < \delta \), then \( |\sqrt{x} - 2| < 0.4 \). ### Graph Explanation: - **Axes**: The graph is plotted on the Cartesian plane with the x-axis labeled as \( x \) and the y-axis labeled as \( y \). - **Function Line**: The curve represents the function \( y = \sqrt{x} \), which steadily increases as \( x \) increases. - **Horizontal Lines**: Two horizontal green lines are drawn at \( y = 2.4 \) and \( y = 1.6 \). These lines are used to represent the bounds within \( |\sqrt{x} - 2| < 0.4 \). - **Vertical Lines**: Two vertical lines intersect the function curve at the points corresponding to \( y = 2.4 \) and \( y = 1.6 \). The x-coordinates of these intersections suggest the values for \( ? \) in question, and you are asked to identify these values related to \( x = 4 \). - **Target Point**: The point on the curve where \( x = 4 \) corresponds to \( \sqrt{4} = 2 \). ### Task: You need to define the distance \( \delta \) around \( x = 4 \) such that all \( x \) within this range (\( |x - 4| < \delta \)) have corresponding \( y \)-values on the graph within the range \( 1.6 < y < 2.4 \). ### Solution Box: - A box labeled \(\delta =\) is provided to input the calculated \( \delta \).
Expert Solution
Step 1

Given Data:

Let us consider the given graph,

Advanced Math homework question answer, step 1, image 1

fx = x if x-4 <δ then x-2<0.4

 

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