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f(x) = [10.5x 1]

f(x) = [10.5x 1]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Graph each function.

Identify the domain and range.

The image presents two mathematical functions and their corresponding graphs. **Function Definitions:** 1. **Function \( f(x) = \lfloor 0.5x \rfloor \):** - This function applies the floor operation to half of the input value \( x \). The floor function \(\lfloor y \rfloor\) returns the greatest integer less than or equal to \( y \). 2. **Function \( g(x) \):** - This is a piecewise function defined as follows: - \( \lfloor x \rfloor \) if \( x < -4 \) - \( x + 1 \) if \( -4 \leq x \leq 5 \) - \(-|x|\) if \( x > 3 \) **Graph Descriptions:** 1. **Graph for \( f(x) = \lfloor 0.5x \rfloor \):** - The graph on the left is a grid with X and Y axes. - It depicts the floor function of half of \( x \). The graph is likely to show step-like, horizontal segments, as the floor function produces constant integer values over intervals. 2. **Graph for \( g(x) \):** - The graph on the right is a grid with X and Y axes. - It represents a piecewise function which consists of three different expressions depending on the value of \( x \). Each segment will differ: - For \( x < -4 \), the floor function of \( x \) will create step-like progressions. - For \( -4 \leq x \leq 5 \), a linear increase is expected because of the expression \( x + 1 \). - For \( x > 3 \), the negated absolute value \(-|x|\) will create a decreasing line moving away from the x-axis. The graphs are drawn on identical XY grids, helping visualize how the function values change across different domains.
Transcribed Image Text:The image presents two mathematical functions and their corresponding graphs. **Function Definitions:** 1. **Function \( f(x) = \lfloor 0.5x \rfloor \):** - This function applies the floor operation to half of the input value \( x \). The floor function \(\lfloor y \rfloor\) returns the greatest integer less than or equal to \( y \). 2. **Function \( g(x) \):** - This is a piecewise function defined as follows: - \( \lfloor x \rfloor \) if \( x < -4 \) - \( x + 1 \) if \( -4 \leq x \leq 5 \) - \(-|x|\) if \( x > 3 \) **Graph Descriptions:** 1. **Graph for \( f(x) = \lfloor 0.5x \rfloor \):** - The graph on the left is a grid with X and Y axes. - It depicts the floor function of half of \( x \). The graph is likely to show step-like, horizontal segments, as the floor function produces constant integer values over intervals. 2. **Graph for \( g(x) \):** - The graph on the right is a grid with X and Y axes. - It represents a piecewise function which consists of three different expressions depending on the value of \( x \). Each segment will differ: - For \( x < -4 \), the floor function of \( x \) will create step-like progressions. - For \( -4 \leq x \leq 5 \), a linear increase is expected because of the expression \( x + 1 \). - For \( x > 3 \), the negated absolute value \(-|x|\) will create a decreasing line moving away from the x-axis. The graphs are drawn on identical XY grids, helping visualize how the function values change across different domains.
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