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Given y = f(x) = |x - 2| + |1 − 2x|-|1 - x. Write y as a piecewise function by showing your work for all 8 cases. Neatness is appreciated.

Given y = f(x) = |x - 2| + |1 − 2x|-|1 - x. Write y as a piecewise function by showing your work for all 8 cases. Neatness is appreciated.

Calculus: Early Transcendentals
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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Given y = f(x) = |x2| + |1 − 2x|-|1 - x. Write y as a piecewise function by showing your work for all 8 cases. Neatness is appreciated.
Transcribed Image Text:Given y = f(x) = |x2| + |1 − 2x|-|1 - x. Write y as a piecewise function by showing your work for all 8 cases. Neatness is appreciated.
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Following up on this question how would you graph function like so as was given in response originally in this thread?

**Graph the Function Carefully Here:** ### Graph Description The image contains a coordinate plane with a grid for graphing functions. The graph shows both the x-axis and y-axis, with the x-axis labeled with integers ranging from -2 to 3 and the y-axis ranging from 0 to 5. ### Instructions Below the graph is a prompt to graph a function. The axes are denoted with “x” and “y” and the grid provides a clear space for plotting points accurately. ### Function Table There is a table located below the graph with two columns labeled "Formula" and "Interval," and one row is available for user input. ### Use on Website This image and format can be utilized by educational websites for teaching students how to graph functions step-by-step, testing their understanding of graphing based on given formulas and intervals, and improving their skills in interpreting and plotting data points on a graph.
Transcribed Image Text:**Graph the Function Carefully Here:** ### Graph Description The image contains a coordinate plane with a grid for graphing functions. The graph shows both the x-axis and y-axis, with the x-axis labeled with integers ranging from -2 to 3 and the y-axis ranging from 0 to 5. ### Instructions Below the graph is a prompt to graph a function. The axes are denoted with “x” and “y” and the grid provides a clear space for plotting points accurately. ### Function Table There is a table located below the graph with two columns labeled "Formula" and "Interval," and one row is available for user input. ### Use on Website This image and format can be utilized by educational websites for teaching students how to graph functions step-by-step, testing their understanding of graphing based on given formulas and intervals, and improving their skills in interpreting and plotting data points on a graph.
**Problem Statement:** Given the function \( y = f(x) = |x - 2| + |1 - 2x| - |1 - x| \). **Task:** Write \( y \) as a piecewise function by examining each case based on the properties of absolute values to take into account all 8 possible cases. Clear and organized presentation of work is encouraged to ensure understanding. **Approach:** 1. **Analyze each absolute value expression:** - \( |x - 2| \) can be rewritten depending on whether \( x \geq 2 \) or \( x < 2 \). - \( |1 - 2x| \) depends on whether \( 2x \geq 1 \) or \( 2x < 1 \) (which simplifies to \( x \geq 0.5 \) or \( x < 0.5 \)). - \( |1 - x| \) changes based on whether \( x \leq 1 \) or \( x > 1 \). 2. **Divide the number line based on critical points:** - Critical points are \( x = 0.5 \), \( x = 1 \), and \( x = 2 \). 3. **Determine expressions for each segment:** - For each segment created by the critical points, redefine the function without absolute values, keeping each condition in mind. 4. **Compile all segments into a piecewise function:** - Ensure continuity and correctness across transitions by showing calculations. The solution needs to be presented in a clean and clear piecewise function, verifying each step to maintain mathematical rigor.
Transcribed Image Text:**Problem Statement:** Given the function \( y = f(x) = |x - 2| + |1 - 2x| - |1 - x| \). **Task:** Write \( y \) as a piecewise function by examining each case based on the properties of absolute values to take into account all 8 possible cases. Clear and organized presentation of work is encouraged to ensure understanding. **Approach:** 1. **Analyze each absolute value expression:** - \( |x - 2| \) can be rewritten depending on whether \( x \geq 2 \) or \( x < 2 \). - \( |1 - 2x| \) depends on whether \( 2x \geq 1 \) or \( 2x < 1 \) (which simplifies to \( x \geq 0.5 \) or \( x < 0.5 \)). - \( |1 - x| \) changes based on whether \( x \leq 1 \) or \( x > 1 \). 2. **Divide the number line based on critical points:** - Critical points are \( x = 0.5 \), \( x = 1 \), and \( x = 2 \). 3. **Determine expressions for each segment:** - For each segment created by the critical points, redefine the function without absolute values, keeping each condition in mind. 4. **Compile all segments into a piecewise function:** - Ensure continuity and correctness across transitions by showing calculations. The solution needs to be presented in a clean and clear piecewise function, verifying each step to maintain mathematical rigor.
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