Problem 5 - Contemporary Precalculus

Question

Deal with the greatest integer function of Example \(7,\) which is given by the equation \(y=[x]\). Compute the following values of the function: $$[6.75]$$

Step-by-Step Solution

Answer

Answer: The greatest integer function of 6.75 is 6.

Step 1: Identify the given value

The value given in this problem is \(6.75\).

Step 2: Determine the greatest integer less than or equal to the given value

As the given value is \(6.75\), we need to find the greatest integer less than or equal to it. The integer part of \(6.75\) is \(6\). Since \(6\) is less than or equal to \(6.75\), the greatest integer less than or equal to \(6.75\) is \(6\).

Step 3: Write the answer

The greatest integer function of \(6.75\) is \([6.75] = 6\).

Key Concepts

Calculating Greatest Integer

Understanding how to compute the greatest integer function is crucial for solving similar problems with ease. The greatest integer function, often represented as \([x]\), essentially asks what is the largest whole number that is still less than or equal to a given number. This number is what we call the 'floor' of \(x\). For example, when you have the number \(6.75\), you need to think about the largest integer that doesn't exceed \(6.75\).
This number would be \(6\), because while \(6.75\) contains fractions, the greatest integer less than it is simply \(6\). Many students make the mistake of rounding up, but remember, you're looking for the integer that's less than or equal to the given number.

Floor Function

The floor function is another way to express determining the greatest integer. It's represented by the notation \(\lfloor x \rfloor\) and serves the same purpose as \([x]\). When you see \(\lfloor 6.75 \rfloor\), it asks you to "floor" the number, meaning find the greatest integer that doesn't go over it.

• This is particularly useful in programming and mathematics whenever you need to handle numbers with decimals and you'd prefer to work with whole numbers instead.
• In equation form, for any real number \(x\), \(\lfloor x \rfloor = n\) is defined where \(n\) is the largest integer ≤ \(x\).

For example, \(\lfloor 6.75 \rfloor = 6\), as this is the largest integer that fits the definition.

Step-by-Step Solutions

Breaking the problem down into simple steps can make it much easier to understand and apply. Here's a detailed look at the process:

• **Identify the Value**: Recognize the number you need to work with, like \(6.75\) in our problem.
• **Determine the Greatest Integer**: Find the integer part of the number. For \(6.75\), you look for what whole number doesn't exceed it, which is \(6\).
• **Write the Final Answer**: Use the notation to express the solution. For our example, write \([6.75] = 6\).

These steps help in building a systematic approach to tackle similar exercises. Practice these steps regularly to get confident at swiftly finding solutions to greatest integer problems.