Problem 77 - Beginning and Intermediate Algebra

Question

Let \(f(x)=[x] .\) Find the following function values. $$f\left(-6 \frac{2}{5}\right)$$

Step-by-Step Solution

Answer

The value of the function \(f(x) = [x]\) when \(x = -6\frac{2}{5}\) is \(-7\).

Step 1: Rewrite the given value as mixed fraction

First, let's rewrite the given value, \(-6\frac{2}{5}\), as mixed fraction. $$ x = -6\frac{2}{5} = -\frac{32}{5} $$

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

Now, we need to find the greatest integer less than or equal to \(-\frac{32}{5}\). Since \(-\frac{32}{5}\) is a negative value, we need to find an integer that is less than or equal to it. Observe that \(-\frac{32}{5}\) lies between \(-6\) and \(-7\), with \(-6\) being closer to it. Therefore, \(-7\) is the greatest integer less than or equal to \(-\frac{32}{5}\).

Step 3: Find the function value

Now, we have that $$ f(x) = [x]=\left[-\frac{32}{5}\right]=-7 $$ Thus, the value of the function \(f(x) = [x]\) when \(x = -6\frac{2}{5}\) is \(-7\).

Key Concepts

Floor Function

The floor function, often denoted by \([x]\), is a mathematical function that takes a real number \(x\) and outputs the greatest integer less than or equal to \(x\). Sometimes, it's also called the greatest integer function. This function is especially useful when dealing with real numbers that we need to approximate to an integer value.

To determine the floor function value, you must locate the largest integer that does not exceed the original number. For example:

• If you have a number like 3.8, the floor function value is 3.
• If you have a number like -2.3, the floor function value is -3.

This rule applies to both positive and negative numbers. For positive values, it's often intuitive, as you simply drop the decimal portion. However, for negative numbers, the floor function moves to the next lower integer. This adjustment is necessary because negative fractions, such as -6.4, are considered as being closer to -7 than to -6. This helps in ensuring the floor function remains consistent with finding the greatest integer that does not exceed the starting number.

Mixed Fractions

Mixed fractions combine a whole number and a fraction, making them a common entity in math problems. A number like \(-6\frac{2}{5}\) is a mixed fraction, where you have -6 as the whole number and \(\frac{2}{5}\) as the fractional part.

When converting a mixed fraction to an improper fraction (where the numerator is greater than the denominator), you multiply the whole number by the denominator and add the numerator. For \(-6\frac{2}{5}\), the conversion goes as follows:

• Take the whole number: -6.
• Multiply by the denominator of the fraction, which is 5: \(-6 \times 5 = -30\).
• Add the numerator: \(-30 + 2 = -32\).

So, \(-6\frac{2}{5}\) becomes \(-\frac{32}{5}\). This conversion is crucial for further calculations, such as using the floor function, especially when it comes to avoiding errors in computation.

Integer Estimation

Estimation of integers from real numbers is a fundamental skill. This involves approximating real numbers to the nearest integer, depending on the context.

In the realm of the floor function, integer estimation is about finding the greatest whole number that is less than or equal to a given real number. Let's break it down with an example: If the number is \(-\frac{32}{5}\), you first compute its approximate value, which is \(-6.4\).

• Because \(-6.4\) lies between \(-6\) and \(-7\), and since the floor function asks for the greatest integer that is non-greater than \(-6.4\), it results in \(-7\).

This estimation technique becomes essential when solving problems that require switching between various forms of number representations. By mastering integer estimation, students can proficiently tackle real-world problems that involve rounding and approximating, improving both their analytical and computational abilities.