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Chapter 2: Problem 32

The greatest integer function \(y=[x]\) is defined as the greatest integer \(\leq x\) (i.e., it rounds \(x\) down to the nearest integer below it). a. What is [2]\(?[2.5] ?[2.9999999] ?\) b. Sketch a graph of the greatest integer function for \(0 \leq x<5 .\) Be sure to indicate whether each endpoint is included or excluded. [Note: A bank employee embezzled hundreds of thousands of dollars by inserting software to round down transactions (such as generating interest on an account) to the nearest cent, and siphoning the round-off differences into his account. He was eventually caught.

Short Answer

Expert verified
\begin{align*}[2] & = 2 \[2.5] & = 2 \[2.9999999] & = 2\text{Graph: horizontal segments from (0,0) to (5,4) with open circles at each integer.}\text{Graph: Horizontal segments at y = 0, 1, 2, 3 and 4 with open circles at 1, 2, 3, 4 and 5.}\right. }

Step by step solution

01

Understanding the Greatest Integer Function

The greatest integer function, also known as the floor function, for any real number x, denoted by \[x\], is defined as the greatest integer less than or equal to x.
02

Evaluating [2]

Here, x is 2. The greatest integer less than or equal to 2 is 2 itself. Therefore, \[ 2 \] = 2.
03

Evaluating [2.5]

Here, x is 2.5. The greatest integer less than or equal to 2.5 is 2. Therefore, \[ 2.5 \] = 2.
04

Evaluating [2.9999999]

Here, x is 2.9999999. The greatest integer less than or equal to 2.9999999 is 2. Therefore, \[ 2.9999999 \] = 2.
05

Sketching the Graph for 0 鈮� x < 5

To graph \[ x \] for 0 鈮� x < 5, we distinguish between included and excluded endpoints. From 0 to just below 1, the graph will be a horizontal line at y = 0 but with an open circle at x = 1. From 1 to just below 2, the graph will be a horizontal line at y = 1 but with an open circle at x = 2. This pattern continues up to x < 5.
06

Illustrate Graph

Draw a horizontal line segment from (0,0) to (1,0), with an open circle at (1,0); from (1,1) to (2,1), with an open circle at (2,1); from (2,2) to (3,2), with an open circle at (3,2); from (3,3) to (4,3), with an open circle at (4,3); and from (4,4) to (5,4), with an open circle at (5,4).

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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, also known as the greatest integer function, is an important mathematical concept. Specifically, the floor function rounds a real number down to the nearest integer less than or equal to it. This is represented by the notation \( \lfloor x \rfloor \).

For example, using the floor function:
  • \( \lfloor 2.7 \rfloor = 2 \)
  • \( \lfloor 3.1 \rfloor = 3 \)
  • \( \lfloor -1.9 \rfloor = -2 \)
When you apply the floor function to established numbers, it helps simplify and standardize the values you work with. It's particularly useful in computer science and finance, especially when rounding values and performing integer-based calculations.
real numbers
Real numbers include all the values that can be found on the number line, encompassing integers, fractions, and irrational numbers. They are fundamental in many areas of mathematics. Real numbers can be positive, negative, or zero. Some common examples include:
  • Integers like -3, 0, 7
  • Fractions like \( \frac{1}{2}\ \)
  • Irrational numbers like \( \pi \ \) and \( \sqrt{2}\ \)
Understanding real numbers is crucial for grasping the floor function since it applies to these types of numbers. For any given real number, the floor function helps you find the integer less than or equal to that number.
graphing functions
Graphing functions is a useful way to visually understand mathematical concepts. For the floor function, graphing shows how the function steps down to the nearest integer. When graphing the floor function:
  • Each horizontal segment represents a range on the x-axis
  • The y-value remains constant until x crosses an integer
  • At each integer, there is a 'jump' down to the next integer value
This step-like appearance is helpful to visualize how the function behaves. Each segment appears as a flat line that suddenly drops at integer points. Understanding these graphs can help you solve problems more effectively.
endpoints in graphs
Endpoints in graphs are critical to understand when dealing with discontinuous functions like the floor function. In these graphs, endpoints indicate whether a value is included or excluded at the boundary of a segment. For the floor function:
  • Open circles indicate excluded points
  • Closed circles indicate included points
For instance, the graph of \( \lfloor x \rfloor \) from \( 0 \) to \( 5 \) has open circles at points like \( 1, 2, 3, 4 \) because these points are the end of each interval and not included in the respective intervals. This helps to clearly distinguish the graph鈥檚 behavior at specific points. Understanding endpoints is essential for accurately interpreting and drawing graphs of mathematical functions.

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Most popular questions from this chapter

Calculate the slope and write an equation for the linear function represented by each of the given tables. a. $$ \begin{array}{cc} \hline x & y \\ \hline 2 & 7.6 \\ 4 & 5.1 \\ \hline \end{array} $$ b. $$ \begin{array}{cc} \hline A & W \\ \hline 5 & 12 \\ 7 & 16 \\ \hline \end{array} $$

a. Write an equation for the line parallel to \(y=2+4 x\) that passes through the point (3,7) . b. Find an equation for the line perpendicular to \(y=2+4 x\) that passes through the point (3,7) .

If a town starts with a population of 63,500 that declines by 700 people each year, construct an equation to model its population size over time. How long would it take for the population to drop to \(53,000 ?\)

The exchange rate that a bank gave for euros in October 2006 was 0.79 euros for \(\$ 1\) U.S. They also charged a constant fee of \(\$ 5\) per transaction. The bank's exchange rate from euros to British pounds was 0.66 pounds for 1 euro, with a transaction fee of 4.1 euros. a. Write a general equation for how many euros you got when changing dollars. Use \(E\) for euros and \(D\) for dollars being exchanged. Draw a graph of \(E\) versus \(D\). b. Would it have made any sense to exchange \(\$ 10\) for euros? c. Find a general expression for the percentage of the total euros converted from dollars that the bank kept for the transaction fee. d. Write a general equation for how many pounds you would get when changing euros. Use \(P\) for British pounds and \(E\) for the euros being exchanged. Draw a graph of \(P\) versus \(E\).

The \(y\) -axis, the \(x\) -axis, the line \(x=6,\) and the line \(y=12\) determine the four sides of a 6 -by- 12 rectangle in the first quadrant (where \(x>0\) and \(y>0\) ) of the \(x y\) plane. Imagine that this rectangle is a pool table. There are pockets at the four corners and at the points (0,6) and (6,6) in the middle of each of the longer sides. When a ball bounces off one of the sides of the table, it obeys the "pool rule": The slope of the path after the bounce is the negative of the slope before the bounce. (Hint: It helps to sketch the pool table on a piece of graph paper first.) a. Your pool ball is at (3,8) . You hit it toward the \(y\) -axis, along the line with slope \(2 .\) i. Where does it hit the \(y\) -axis? ii. If the ball is hit hard enough, where does it hit the side of the table next? And after that? And after that? iii. Show that the ball ultimately returns to \((3,8) .\) Would it do this if the slope had been different from \(2 ?\) What is special about the slope 2 for this table? b. A ball at (3,8) is hit toward the \(y\) -axis and bounces off it at \(\left(0, \frac{16}{3}\right) .\) Does it end up in one of the pockets? If so, what are the coordinates of that pocket? c. Your pool ball is at (2,9) . You want to shoot it into the pocket at \((6,0) .\) Unfortunately, there is another ball at (4,4.5) that may be in the way. i. Can you shoot directly into the pocket at (6,0)\(?\) ii. You want to get around the other ball by bouncing yours off the \(y\) -axis. If you hit the \(y\) -axis at \((0,7),\) do you end up in the pocket? Where do you hit the line \(x=6 ?\) iii. If bouncing off the \(y\) -axis at (0,7) didn't work, perhaps there is some point \((0, b)\) on the \(y\) -axis from which the ball would bounce into the pocket at (6,0) Try to find that point.
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