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Explain how to graph the given piecewise-defined function. Be sure to specify the type of endpoint each piece of the function will have and why. f(x) = -x+ 3, x < 2 3, 4-2x, 25x<4 x 24

Explain how to graph the given piecewise-defined function. Be sure to specify the type of endpoint each piece of the function will have and why. f(x) = -x+ 3, x < 2 3, 4-2x, 25x<4 x 24

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**How to Graph a Piecewise-Defined Function** To graph the piecewise-defined function \( f(x) \), follow these steps: 1. **Identify the Pieces:** - The function has three pieces: - \( f(x) = -x + 3 \) for \( x < 2 \) - \( f(x) = 3 \) for \( 2 \leq x < 4 \) - \( f(x) = 4 - 2x \) for \( x \geq 4 \) 2. **Graph Each Piece:** - **First Piece:** \( f(x) = -x + 3 \) - This is a linear function with a slope of -1 and y-intercept at 3. - Graph this line for \( x < 2 \). - Since the inequality is strict (\( x < 2 \)), use an open circle at the point where \( x = 2 \). - **Second Piece:** \( f(x) = 3 \) - This is a horizontal line at \( y = 3 \). - Graph this line segment from \( x = 2 \) to \( x = 4 \). - Since \( 2 \leq x \), use a closed circle at \( x = 2 \). - Since \( x < 4 \), use an open circle at \( x = 4 \). - **Third Piece:** \( f(x) = 4 - 2x \) - This is a linear function with a slope of -2 and a y-intercept at 4. - Graph this line for \( x \geq 4 \). - Since the inequality is non-strict (\( x \geq 4 \)), use a closed circle at the point where \( x = 4 \). 3. **Combine the Graph:** - Plot each line segment on the same set of axes. - Ensure the endpoints are clearly marked according to the inequality signs. 4. **Understanding Endpoints:** - The type of endpoint (open or closed circle) represents whether the endpoint value is included in the graph. Closed circles indicate inclusion (\( \leq \) or \( \geq \)), and open circles indicate exclusion (\( < \) or \( > \)). By carefully examining each part of the
Transcribed Image Text:**How to Graph a Piecewise-Defined Function** To graph the piecewise-defined function \( f(x) \), follow these steps: 1. **Identify the Pieces:** - The function has three pieces: - \( f(x) = -x + 3 \) for \( x < 2 \) - \( f(x) = 3 \) for \( 2 \leq x < 4 \) - \( f(x) = 4 - 2x \) for \( x \geq 4 \) 2. **Graph Each Piece:** - **First Piece:** \( f(x) = -x + 3 \) - This is a linear function with a slope of -1 and y-intercept at 3. - Graph this line for \( x < 2 \). - Since the inequality is strict (\( x < 2 \)), use an open circle at the point where \( x = 2 \). - **Second Piece:** \( f(x) = 3 \) - This is a horizontal line at \( y = 3 \). - Graph this line segment from \( x = 2 \) to \( x = 4 \). - Since \( 2 \leq x \), use a closed circle at \( x = 2 \). - Since \( x < 4 \), use an open circle at \( x = 4 \). - **Third Piece:** \( f(x) = 4 - 2x \) - This is a linear function with a slope of -2 and a y-intercept at 4. - Graph this line for \( x \geq 4 \). - Since the inequality is non-strict (\( x \geq 4 \)), use a closed circle at the point where \( x = 4 \). 3. **Combine the Graph:** - Plot each line segment on the same set of axes. - Ensure the endpoints are clearly marked according to the inequality signs. 4. **Understanding Endpoints:** - The type of endpoint (open or closed circle) represents whether the endpoint value is included in the graph. Closed circles indicate inclusion (\( \leq \) or \( \geq \)), and open circles indicate exclusion (\( < \) or \( > \)). By carefully examining each part of the
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