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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![### Graph the Function
Consider the function defined as:
\[ f(x) = \lfloor x \rfloor + 1 \]
Here, \(\lfloor x \rfloor\) represents the floor function, which returns the greatest integer less than or equal to \(x\).
#### Steps to Graph the Function:
1. Begin with an \(xy\)-coordinate plane.
2. For each integer value of \(x\), plot the point corresponding to that value of \(x\), but adjust the \(y\)-value by adding 1 to the floor function result.
#### Explanation:
- For \( x \) in the interval \([0, 1)\):
- \(\lfloor x \rfloor\) is 0.
- Thus, \( f(x) = 0 + 1 = 1 \).
- Plot a horizontal line segment from \((0,1)\) to \((1,1)\), excluding the point \((1,1)\) but including the point \((0,1)\).
- For \( x \) in the interval \([1, 2)\):
- \(\lfloor x \rfloor\) is 1.
- Thus, \( f(x) = 1 + 1 = 2 \).
- Plot a horizontal line segment from \((1,2)\) to \((2,2)\), excluding the point \((2,2)\) but including the point \((1,2)\).
- Continue this process for all integer intervals.
#### Graph Features:
- Each segment will be 1 unit above the integer value of the floor function.
- The graph will consist of horizontal line segments, each extending from left-closed and right-open within the interval.
- At each new integer value of \(x\), there will be a jump discontinuity—the graph "jumps" vertically by 1 unit.
#### Coordinate Plane:
- The \(x\)-axis represents the independent variable \(x\).
- The \(y\)-axis represents the dependent variable \(f(x)\).
- The axes should be labeled with appropriate scales and tick marks.
This detailed explanation and the introduction to the floor function, along with the segmentation of intervals, help in understanding how to draw and interpret the graph of the given function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F71b73f36-898d-4ba7-a9cf-414d54176493%2F6fj3dnj_processed.png&w=3840&q=75)
Transcribed Image Text:### Graph the Function
Consider the function defined as:
\[ f(x) = \lfloor x \rfloor + 1 \]
Here, \(\lfloor x \rfloor\) represents the floor function, which returns the greatest integer less than or equal to \(x\).
#### Steps to Graph the Function:
1. Begin with an \(xy\)-coordinate plane.
2. For each integer value of \(x\), plot the point corresponding to that value of \(x\), but adjust the \(y\)-value by adding 1 to the floor function result.
#### Explanation:
- For \( x \) in the interval \([0, 1)\):
- \(\lfloor x \rfloor\) is 0.
- Thus, \( f(x) = 0 + 1 = 1 \).
- Plot a horizontal line segment from \((0,1)\) to \((1,1)\), excluding the point \((1,1)\) but including the point \((0,1)\).
- For \( x \) in the interval \([1, 2)\):
- \(\lfloor x \rfloor\) is 1.
- Thus, \( f(x) = 1 + 1 = 2 \).
- Plot a horizontal line segment from \((1,2)\) to \((2,2)\), excluding the point \((2,2)\) but including the point \((1,2)\).
- Continue this process for all integer intervals.
#### Graph Features:
- Each segment will be 1 unit above the integer value of the floor function.
- The graph will consist of horizontal line segments, each extending from left-closed and right-open within the interval.
- At each new integer value of \(x\), there will be a jump discontinuity—the graph "jumps" vertically by 1 unit.
#### Coordinate Plane:
- The \(x\)-axis represents the independent variable \(x\).
- The \(y\)-axis represents the dependent variable \(f(x)\).
- The axes should be labeled with appropriate scales and tick marks.
This detailed explanation and the introduction to the floor function, along with the segmentation of intervals, help in understanding how to draw and interpret the graph of the given function.
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