Evaluate F(7.6) Given F(x) = (1/2) Floor(x)

Introduction

In this article, we will delve into the concept of the floor function and apply it to evaluate a specific function. The floor function, denoted by $\lfloor x \rfloor$, is a fundamental concept in mathematics that returns the greatest integer less than or equal to $x$. Our task is to determine the value of $f(7.6)$ given that $f(x) = \frac{1}{2}\lfloor x\rfloor$. This problem combines the understanding of the floor function with basic arithmetic operations, making it a valuable exercise in mathematical reasoning. Let's break down the problem step by step to arrive at the correct solution.

Understanding the Floor Function

The floor function, often denoted as $\lfloor x \rfloor$, plays a crucial role in various mathematical contexts, including number theory, computer science, and real analysis. At its core, the floor function provides a way to round down a real number to the nearest integer. This function maps a real number $x$ to the greatest integer that is less than or equal to $x$. In simpler terms, it essentially "chops off" the decimal part of the number while keeping the integer part. For example, $\lfloor 3.14 \rfloor = 3$, $\lfloor 5.99 \rfloor = 5$, and $\lfloor -2.3 \rfloor = -3$. It's important to note that for negative numbers, the floor function rounds down to the next lower integer, which is why $\lfloor -2.3 \rfloor$ equals $-3$ rather than $-2$. The floor function has numerous applications. In computer science, it is used in algorithms that require integer values, such as array indexing and memory allocation. In number theory, it appears in various formulas and theorems related to divisibility and modular arithmetic. Understanding the behavior of the floor function is essential for solving problems that involve non-integer values and integer constraints. This concept is also foundational for understanding more advanced mathematical concepts like limits and continuity in calculus. Mastering the floor function provides a strong basis for tackling more complex mathematical problems.

Evaluating f(7.6)

Now that we have a solid understanding of the floor function, we can proceed to evaluate $f(7.6)$ given the function definition $f(x) = \frac{1}{2}\lfloor x\rfloor$. The first step is to determine the value of the floor function applied to $7.6$, which is $\lfloor 7.6 \rfloor$. According to the definition of the floor function, we need to find the greatest integer less than or equal to $7.6$. In this case, the greatest integer less than or equal to $7.6$ is $7$. Therefore, $\lfloor 7.6 \rfloor = 7$. Next, we substitute this value into the function $f(x)$. So, $f(7.6) = \frac{1}{2} \times \lfloor 7.6 \rfloor = \frac{1}{2} \times 7$. Performing this multiplication, we get $f(7.6) = \frac{7}{2}$, which is equal to $3.5$. Thus, the value of $f(7.6)$ is $3.5$. This straightforward calculation demonstrates how the floor function works in conjunction with other arithmetic operations. By first applying the floor function to obtain an integer value and then performing the multiplication, we arrive at the final result. This type of evaluation is common in various mathematical contexts and highlights the importance of understanding the properties and behavior of different functions. The precise application of the floor function ensures that we obtain the correct result, reinforcing the significance of meticulous mathematical operations.

Detailed Solution

To find the value of $f(7.6)$, where $f(x) = \frac{1}{2}\lfloor x\rfloor$, we need to follow a specific step-by-step approach. First, we must evaluate the floor function for $x = 7.6$. The floor function, denoted as $\lfloor x \rfloor$, gives the greatest integer less than or equal to $x$. In this case, $\lfloor 7.6 \rfloor$ means we need to find the largest integer that is less than or equal to $7.6$. The integers around $7.6$ are $7$ and $8$. Since $7$ is the greatest integer that is less than or equal to $7.6$, we have $\lfloor 7.6 \rfloor = 7$. Now that we have the value of the floor function, we can substitute it into the expression for $f(x)$. We have $f(7.6) = \frac{1}{2}\lfloor 7.6\rfloor$. Substituting the value we found, we get $f(7.6) = \frac{1}{2} \times 7$. Performing the multiplication, we have $f(7.6) = \frac{7}{2}$. To express this as a decimal, we divide $7$ by $2$, which gives us $3.5$. Therefore, $f(7.6) = 3.5$. This step-by-step solution clearly demonstrates the process of applying the floor function and then using the result in the given function $f(x)$. Understanding each step ensures that we arrive at the correct answer, and it highlights the importance of following the order of operations in mathematical problems. By breaking down the problem into smaller, manageable parts, we can confidently solve it and gain a deeper understanding of the underlying mathematical concepts.

Conclusion

In conclusion, to find $f(7.6)$ given the function $f(x) = \frac{1}{2}\lfloor x\rfloor$, we first evaluated the floor function of $7.6$, which is $\lfloor 7.6 \rfloor = 7$. Then, we substituted this value into the function to get $f(7.6) = \frac{1}{2} \times 7 = 3.5$. Therefore, the correct answer is $3.5$. Understanding the floor function and its application is crucial for solving such problems accurately. This exercise not only reinforces the concept of the floor function but also highlights the importance of step-by-step problem-solving in mathematics. By breaking down the problem into manageable parts, we can systematically arrive at the correct solution. The floor function, as demonstrated, plays a vital role in mapping real numbers to integers, which is a fundamental operation in various mathematical and computational contexts. The final answer of $3.5$ illustrates the combined effect of the floor function and basic arithmetic, showcasing how different mathematical concepts work together to produce a solution.

Final Answer

The final answer is B. 3.5.