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Chapter 10: Problem 11

Throughout this lesson you have used the greatest integer function \(y=[x] .\) Use your graphing calculator to view the graph of this function and find the \(y\) -coordinate for each of the following \(x\) -values. a. 2.3 b. 2.99 c. 3 d. 3.01 e. 3.99

Short Answer

Expert verified
\([2.3]=2, [2.99]=2, [3]=3, [3.01]=3, [3.99]=3\)

Step by step solution

01

Understanding the Greatest Integer Function

The greatest integer function, also known as the floor function, maps \(x\) to the largest integer less than or equal to \(x\). If a number is an integer, the function returns that very number, but if the number is a decimal, the function returns the integer component of that number. For instance, \([2.3]\) would return 2 because 2 is the largest integer not greater than 2.3.
02

Evaluate for 2.3

Consider \(x=2.3\). The greatest integer that is less than or equal to 2.3 is 2. So, \([2.3]=2\).
03

Evaluate for 2.99

Consider \(x=2.99\). The greatest integer that is less than or equal to 2.99 is 2. So, \([2.99]=2\).
04

Evaluate for 3

Consider \(x=3\). The greatest integer that is less than or equal to 3 is 3 itself. So, \([3]=3\).
05

Evaluate for 3.01

Consider \(x=3.01\). The greatest integer that is less than or equal to 3.01 is 3. So, \([3.01]=3\).
06

Evaluate for 3.99

Consider \(x=3.99\). The greatest integer that is less than or equal to 3.99 is 3. So, \([3.99]=3\).

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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, or greatest integer function, is quite intriguing! It takes any real number and maps it to the largest integer that is less than or equal to it. Think of it as a function that "floors" or rounds down any number.

For example, when you have a number like 2.3, the floor function will return 2, because 2 is the biggest integer that doesn鈥檛 exceed 2.3. Similarly, for a number like 3.99, the function returns 3, as it is the largest integer below 3.99.
  • This function is helpful in scenarios where you need whole numbers out of decimals.
  • It鈥檚 used in computing, mathematics, and various other fields that require integer outputs.
Keep in mind that if the input is already an integer, the floor function just returns the same value. Isn't that neat? It's like having a favorite shortcut for dealing with numbers!
graphing calculator
Using a graphing calculator can bring functions to life. It's a practical tool for visualizing mathematical concepts such as the floor function.

To check out the graph of a greatest integer function, plug it into your graphing calculator. You鈥檒l notice a step-like graph, which shows the function mapping each input to its floor value. Each segment is flat because the output remains constant until the next integer is reached.
  • The steps occur at integer values of x.
  • This tool is great for seeing the behavior of functions over different ranges.
By using the calculator, you can effectively see how the function behaves at decimal and whole number inputs alike. This makes learning much more interactive and engaging.
integer components
Every decimal number can be split into two parts: an integer and a fractional component.

In the context of the floor function, our center of attention is the integer component. This is the number that the floor function retrieves from a given input.
  • For 5.67, the integer component is 5.
  • For 10.01, it's 10.
The floor function essentially ignores the fractional part, making the integer component the hero of the operation. This principle is vital when rounding numbers down to the nearest integer is required.

Recognizing these integer components is crucial in math since so many operations and functions hinge on them.
evaluating functions
When evaluating functions like the floor function, we focus on applying a rule or operation to a set of inputs to obtain their respective outputs.

Let's break it down with an example: for evaluating the greatest integer function at an input of 2.3, we apply the floor function rule, obtaining the output 2.
  • Another example: for 2.99, we still get 2 as the output.
  • For exact integers like 3, the function simply returns 3.
Evaluating functions involves understanding the role of the function鈥檚 rule in modifying inputs, which is essential for mastering various mathematical operations.

Through consistent practice, evaluating functions becomes second nature, and you'll gain a clearer understanding of how different functions affect numbers.

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Most popular questions from this chapter

Home heating oil is sold by the gallon. Last winter, the Romano family used 370 gallons of oil at a price of \(\$ 3.91\) per gallon. If the price increases 9\(\%\) next year, what will their approximate heating expense be? Round to the nearest ten dollars.

Under his household expense budget category, Mark has allocated \(\$ 60\) per month for pet food. He can purchase wet food in a can for \(\$ 1.50\) per can or dry food in a bag for \(\$ 3\) per bag. a. Determine a budget line equation for this situation. b. Graph the budget line that will depict the different combinations of cans and bags that Mark can purchase while still remaining within his budget. c. Name a combination of bags and cans that will allow him to d. Name a combination of bags and cans that will keep him under budget. e. Name a combination of bags and cans that will cause him to be over budget.

Interpret the quote in the context of what you know about the major issues in energy consumption today.

Text-Time charges \(\$ 25\) for a texting plan with 300 text messages included. If the customer goes over the 300 messages, the cost is \(\$ 0.10\) per message. They have an unlimited plan for \(\$ 48\) per month. a. If \(x\) represents the number of text messages, and \(c(x)\) represents the cost of the messages, express \(c(x)\) as a piecewise function. b. Graph the function from part a. c. What are the coordinates of the cusp in your graph from part b? d. On the same axes as your graph from part b, graph the function \(c(x)=48,\) which represents the cost under the unlimited plan. e. For what number of text messages are the costs of the two different plans the same?

The Zwerling family installed central air conditioning in their house this summer. They are compring the electric bills of this summer and last summer. The data is shown. $$\begin{array}{|c|c|c|}\hline \text { Month } & {\text { This Summer }} & {\text { Last Summer }} \\ \hline \text { June } & {\$ 311.20} & {\$ 179.90} \\\ {\text { July }} & {300.65} & {\$ 203.40} \\ {\text { August }} & {302.50} & {\$ 201.11}\end{array}$$ a. What was the total electric bill this summer? b. What was the total electric bill last summer? c. Did the bill increase more or less than 50\(\%\) ?
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