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Complete the description of the piecewise function graphed below. 0 9 -7 -6 -5 -4 -3 -2 -1 H -2- 9 { f(x) = { { 3- 2 1 -4 -5- --6- 2 13 12 5 6 q if -6 ≤ x ≤-3 if -3 < x < 2 if 2 < x < 6

Complete the description of the piecewise function graphed below. 0 9 -7 -6 -5 -4 -3 -2 -1 H -2- 9 { f(x) = { { 3- 2 1 -4 -5- --6- 2 13 12 5 6 q if -6 ≤ x ≤-3 if -3 < x < 2 if 2 < x < 6

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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### Identifying the Piecewise Function To fully describe the piecewise function shown in the graph, we need to examine each segment separately and determine the functions for the specified intervals. #### Graph Analysis - **First Segment (-6 <= x <= -3)**: - This segment is represented by a line that starts at the point (-6, -5) and ends at the point (-3, 1). - To find the equation of this line, use the two-point form of the line equation: \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \). - Substituting the given points (-6, -5) and (-3, 1), the slope (m) is \( \frac{1 - (-5)}{-3 - (-6)} = \frac{6}{3} = 2 \). - The line equation is \( y + 5 = 2(x + 6) \), so simplify it to get \( y = 2x + 7 \). - **Second Segment (-3 < x <= 2)**: - This segment is a constant function that starts at the point (-3, 1) and ends at the point (2, -3). - There is no variation in y, hence, the function is constant \( y = -3 \). - **Third Segment (2 < x <= 6)**: - This segment is represented by a line that starts at the point (2, -3) and ends at the point (6, 5). - To find the equation of this line, use the two-point form of the line equation. - The slope (m) is \( \frac{5 - (-3)}{6 - 2} = \frac{8}{4} = 2 \). - The line equation is \( y + 3 = 2(x - 2) \), so simplify it to get \( y = 2x - 7 \). #### Completed Function Based on the analysis, the piecewise function \( f(x) \) is: \[ f(x) = \begin{cases} 2x + 7 & \text{if } -6 \leq x \leq -3 \\ -3 & \text{if } -3 <
Transcribed Image Text:### Identifying the Piecewise Function To fully describe the piecewise function shown in the graph, we need to examine each segment separately and determine the functions for the specified intervals. #### Graph Analysis - **First Segment (-6 <= x <= -3)**: - This segment is represented by a line that starts at the point (-6, -5) and ends at the point (-3, 1). - To find the equation of this line, use the two-point form of the line equation: \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \). - Substituting the given points (-6, -5) and (-3, 1), the slope (m) is \( \frac{1 - (-5)}{-3 - (-6)} = \frac{6}{3} = 2 \). - The line equation is \( y + 5 = 2(x + 6) \), so simplify it to get \( y = 2x + 7 \). - **Second Segment (-3 < x <= 2)**: - This segment is a constant function that starts at the point (-3, 1) and ends at the point (2, -3). - There is no variation in y, hence, the function is constant \( y = -3 \). - **Third Segment (2 < x <= 6)**: - This segment is represented by a line that starts at the point (2, -3) and ends at the point (6, 5). - To find the equation of this line, use the two-point form of the line equation. - The slope (m) is \( \frac{5 - (-3)}{6 - 2} = \frac{8}{4} = 2 \). - The line equation is \( y + 3 = 2(x - 2) \), so simplify it to get \( y = 2x - 7 \). #### Completed Function Based on the analysis, the piecewise function \( f(x) \) is: \[ f(x) = \begin{cases} 2x + 7 & \text{if } -6 \leq x \leq -3 \\ -3 & \text{if } -3 <
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