Problem 103 - Applied Calculus

Question

For any \(x\), the function \(\operatorname{INT}(x)\) is defined as the greatest integer less than or equal to \(x\). For example, \(\operatorname{INT}(3.7)=3\) and \(\operatorname{INT}(-4.2)=-5\) a. Use a graphing calculator to graph the function \(y_{1}=\operatorname{INT}(x) .\) (You may need to graph it in DOT mode to eliminate false connecting lines.) b. From your graph, what are the domain and range of this function?

Step-by-Step Solution

Answer

The domain is all real numbers; the range is all integers.

Step 1: Understanding the INT Function

The function \(\operatorname{INT}(x)\) gives the greatest integer that is less than or equal to \(x\). It essentially 'floors' the value of \(x\), moving it to the nearest integer less than or equal.

Step 2: Graphing the INT Function

To graph the function \(y_1 = \operatorname{INT}(x)\), use your graphing calculator and input the function. Make sure the calculator is in DOT mode so that the graph appears as distinct steps rather than a continuous line, since the INT function is a step function.

Step 3: Identifying the Domain of the INT Function

The domain of the function \(y_1 = \operatorname{INT}(x)\) consists of all real numbers. This is because for any real number \(x\), there is always a greatest integer less than or equal to that number. Therefore, the domain is \((-\infty, \infty)\).

Step 4: Identifying the Range of the INT Function

The range of the function consists of all integers because for any real number \(x\), the output of \(\operatorname{INT}(x)\) is an integer. Therefore, the range is the set of all integers, usually denoted as \(\mathbb{Z}\).

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps us visually understand mathematical functions, including the step function known as \(\operatorname{INT}(x)\). To graph this function, you'll want to input \(y_1=\operatorname{INT}(x)\). But what's unique here is that the graph of \(\operatorname{INT}(x)\) doesn't form a continuous line, it creates a series of flat horizontal steps.

• Each step represents a range of values for \(x\) that produce the same greatest integer result.
• When using the calculator, switch to DOT mode. This is crucial because DOT mode prevents the calculator from connecting these steps with lines, which would misrepresent the graph of the step function.

Understanding how to adjust settings on your graphing calculator is essential for accurately representing different types of functions, especially those with unique characteristics like the INT function.

Domain and Range

The concepts of domain and range are fundamental when dealing with functions, including the greatest integer function \(\operatorname{INT}(x)\).

• Domain: For \(y_1=\operatorname{INT}(x)\), the domain is all real numbers \(\mathbb{R}\). This means you can input any real number into the function, and it will produce a valid output.
• Range: The range of this function is the set of all integers \(\mathbb{Z}\). Despite the infinite possibilities of \(x\), \(\operatorname{INT}(x)\) will always map these to an integer, thus making its range discrete.

In more visual terms, on the graph, you'd see this function jumping from one integer value to another, creating a step-like appearance. Hence, understanding the domain and range gives clarity on the inputs the function accepts and the variety of outputs it can produce.

Greatest Integer Function

The greatest integer function, denoted as \(\operatorname{INT}(x)\), is pivotal in a group of functions known as step functions. Here's how it operates:

• This function captures the greatest integer less than or equal to any given real number \(x\).
• For example, \(\operatorname{INT}(3.7) = 3\), which means it drops down to the largest whole number below 3.7.
• It also applies to negative numbers: \(\operatorname{INT}(-4.2) = -5\), stepping further down to the nearest integer.

The significance of this function is evident in its diverse applications, especially in computer science and economics, where rounding down to the nearest integer has practical importance. This mathematical floor operation makes it unique, as it handles values by shifting them into a series of integers, constructing a staircase-like arrangement on a graph.