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Chapter 12: Problem 84

Graph the following greatest integer functions. $$k(x)=[x]+3$$

Short Answer

Expert verified
To graph \(k(x)=[x]+3\), plot the horizontal line segments on the intervals: - \([-1,0)\) with a y-value of 3. - \([0,1)\) with a y-value of 4. - \([1,2)\) with a y-value of 5. - \([2,3)\) with a y-value of 6. Include a closed circle at the beginning of each segment (\(x=n\)) and an open circle at the end (\(x=n+1\)).

Step by step solution

01

Understand the graph of the greatest integer function [x]

The graph of the greatest integer function, also known as the floor function [x], consists of a series of consecutive horizontal line segments. Each line segment corresponds to a domain (range of x-values) of the form [n, n+1), where n is an integer. The y-value of each piecewise horizontal line segment is equal to the greatest integer less than or equal to x.
02

Determine the graph of k(x) = [x] + 3

To find the graph of \(k(x) = [x] + 3\), we simply take the graph of [x] determined in step 1 and add 3 to the y-value of each line segment, thus shifting the line segments vertically upward by 3 units to find the corresponding \(k(x)\) value for each x.
03

Graph the horizontal line segments

We will begin by drawing horizontal line segments from -3 to 3 to represent the range of x-values we are graphing. For each integer n, draw a line segment on the interval [n, n+1) with a y-value of [n] + 3. The graph of \(k(x) = [x] + 3\) will have horizontal line segments on the intervals: - \([-1,0)\) with a y-value of 3 (since \([x] = -1\) for x in the interval \([-1,0)\)). - \([0, 1)\) with a y-value of 4 (since \([x] = 0\) for x in the interval \([0, 1)\)). - \([1, 2)\) with a y-value of 5 (since \([x] = 1\) for x in the interval \([1, 2)\)). - \([2, 3)\) with a y-value of 6 (since \([x] = 2\) for x in the interval \([2, 3)\)). Make sure to draw an open circle (hollow dot) at the end of each segment where the segment is not included (\(x = n + 1\)), and a closed circle (filled dot) at the beginning of each segment where the segment is included (\(x = n\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Floor Function
The floor function is a mathematical function that maps a real number to the greatest integer less than or equal to that number. It is often denoted by \([x]\). This function rounds down any decimal number to its nearest whole integer. For example, \([2.9] = 2\) and \([-1.5] = -2\).

The floor function is a type of step function, as the values jump from one integer to the next. For each interval \([n, n+1)\) where \(n\) is an integer, the function returns \(n\) as its output. This creates distinctive horizontal line segments on a graph, which are characteristic of the floor function. These segments do not connect in a continuous path, making the floor function quite different from typical linear functions.
Graphing Functions
Graphing a function involves plotting its values (outputs) on a coordinate grid based on given inputs (x-values). When graphing the greatest integer function \([x]\), you will see a series of steps formed by the horizontal segments.

Each segment corresponds to a specific interval on the x-axis. For example, for \([x]+3\), we shift each horizontal segment of \([x]\) up by 3 units. This means that for every x-value in a particular step, we add 3 to the greatest integer function to determine the y-value at which to plot each segment. Using the example of \(k(x) = [x] + 3\), we see segments appearing higher in the grid due to this vertical shift.
Piecewise Functions
Piecewise functions are functions that have different definitions depending on which part of the domain you are evaluating. They are represented by multiple sub-functions, each applying to a specific interval of the independent variable, usually x.

The greatest integer function \([x]\) can be considered a piecewise function, as it consists of distinct constant values for each interval \([n, n+1)\).

### Example of a Piecewise Function:- For any \(x\) in \([-1, 0)\), the value is -1.- For \(x\) in \([0, 1)\), the value is 0.
- And so on for other intervals.

Aligning with piecewise function rules, \([x]+3\) will maintain these constant pieces but offset them upward by 3 units.
Horizontal Line Segments
Horizontal line segments are straight, flat lines that run left to right. They have no slope because the y-value remains the same over different x-values.

In the context of the greatest integer function like \([x]\), each horizontal segment represents a constant y-value over a unique interval \([n, n+1)\). This unique property causes the graphs to have a jump or step-like appearance, which makes it easy to identify.

When shifting these horizontal line segments, such as with the function \([x]+3\), each segment is just moved upwards but remains horizontal. These horizontal segments are important because they visualize how the function behaves over specific x-ranges with precision.

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Most popular questions from this chapter

Use the transformation techniques to graph each of the following functions. $$y=-\sqrt{x+2}$$

Graph \(f(x)=\sqrt[3]{x}\) by plotting points. (Hint: Make a table of values and choose \(0,\) positive, and negative numbers for \(x\) ) Then, use the transformation techniques discussed in this section to graph each of the following functions. $$\text{a)}\quad g(x)=\sqrt[3]{x}+4\quad\text{b)}\quad h(x)=-\sqrt[3]{x}\quad\text{c)} \quad k(x)=\sqrt[3]{x-2}\quad\text{d)}\quad $r(x)=-\sqrt[3]{x}-3$$

Graph the following greatest integer functions. $$h(x)=[x-1]$$

The area, \(A,\) of a circle is a function of its radius, \(r\) A. Write an equation using function notation to describe this relationship between \(A\) and \(r\) B. If the radius is given in centimeters, find \(A(3)\) and explain what this means in the context of the problem. C. If the radius is given in inches, find \(A(5)\) and explain what this means in the context of the problem. D. What is the radius of a circle with an area of \(64 \pi\) in \(^{2} ?\)

Graph the following piecewise functions. $$k(x)=\left\\{\begin{array}{cc}x+1, & x \geq-2 \\\2 x+8, & x<-2\end{array}\right.$$
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