Evaluate G(x) = 2⌊x⌋ - 1 At X = -2.3

Let's dive into evaluating the function $g(x) = 2\lfloor x \rfloor - 1$ at the specific point $x = -2.3$. This involves understanding the floor function, denoted by $\lfloor x \rfloor$, which returns the greatest integer less than or equal to $x$. It's a fundamental concept in mathematics, especially when dealing with real numbers and their integer parts. The floor function essentially 'rounds down' a number to the nearest integer. For positive numbers, this is straightforward, but for negative numbers, it's important to remember that 'rounding down' means moving to the next lower integer, which is more negative. We'll carefully apply this concept to find the correct value of $g(-2.3)$. Understanding how to evaluate functions, particularly those involving floor or ceiling functions, is crucial for various mathematical applications, including computer science, number theory, and real analysis. The process helps clarify how mathematical functions operate on different types of input and produce specific outputs based on defined rules. In our case, the combination of the floor function with a linear expression adds an interesting layer to the evaluation, making it a good example to demonstrate the importance of precise calculation and conceptual understanding in mathematics. Remember guys, math can be fun and rewarding, especially when we break down complex problems into smaller, manageable steps!

Understanding the Floor Function

The floor function, denoted as $\lfloor x \rfloor$, might seem simple, but it's a cornerstone in various mathematical contexts. The floor function takes a real number, $x$, as input and returns the greatest integer that is less than or equal to $x$. Essentially, it rounds the number down to the nearest integer. For instance, $\lfloor 3.14 \rfloor = 3$, $\lfloor 5 \rfloor = 5$, and $\lfloor -2.7 \rfloor = -3$. The key here is the "less than or equal to" part of the definition. This means that for positive integers and zero, the floor function simply returns the integer itself. However, for negative numbers, it's important to remember the direction of the number line. When we round a negative number down, we're moving towards more negative values. The floor function is widely used in computer science, particularly in algorithms that require integer arithmetic or discrete calculations. For example, it's used in array indexing, memory allocation, and various numerical methods. In mathematics, the floor function appears in number theory, real analysis, and combinatorics, often in the context of counting problems or defining piecewise functions. Understanding the floor function is also essential for grasping the behavior of other related functions like the ceiling function ($\lceil x \rceil$), which rounds a number up to the nearest integer, and the fractional part function ({$x$} = $x - \lfloor x \rfloor$), which gives the decimal part of a number. So, next time you encounter the floor function, remember it's all about finding the greatest integer below or equal to the given number. And remember, mastering these fundamental concepts is what makes you a math whiz!

Evaluating $g(-2.3)$

Now, let's apply this understanding to evaluate $g(-2.3)$, where $g(x) = 2\lfloor x \rfloor - 1$. First, we need to find the value of $\lfloor -2.3 \rfloor$. Remember, the floor function rounds down to the nearest integer. In this case, $-2.3$ lies between $-3$ and $-2$ on the number line. Rounding down means we go to $-3$. Therefore, $\lfloor -2.3 \rfloor = -3$. Now that we have the value of the floor function at $x = -2.3$, we can substitute it into the expression for $g(x)$. So, $g(-2.3) = 2 \cdot (-3) - 1$. Multiplying $2$ by $-3$ gives us $-6$. Then, subtracting $1$ from $-6$ results in $-7$. Therefore, $g(-2.3) = -7$. This calculation shows how important it is to correctly apply the definition of the floor function, especially when dealing with negative numbers. A common mistake is to round $-2.3$ to $-2$, which would lead to an incorrect result. Always visualize the number line and remember that rounding down means moving to the next lower integer, regardless of whether the number is positive or negative. Evaluating functions like this is a fundamental skill in mathematics and is essential for solving more complex problems in calculus, analysis, and other areas. By carefully applying the definition of the floor function and following the order of operations, we can confidently determine the value of $g(-2.3)$.

Step-by-Step Solution

To make it crystal clear, let's break down the solution into a step-by-step process:

1. Identify the Function: We are given the function $g(x) = 2\lfloor x \rfloor - 1$.
2. Identify the Input: We need to evaluate the function at $x = -2.3$.
3. Evaluate the Floor Function: Find $\lfloor -2.3 \rfloor$. Since the floor function rounds down to the nearest integer, $\lfloor -2.3 \rfloor = -3$.
4. Substitute into the Function: Substitute the value of the floor function into the expression for $g(x)$: $g(-2.3) = 2 \cdot (-3) - 1$.
5. Perform the Multiplication: Multiply $2$ by $-3$: $2 \cdot (-3) = -6$.
6. Perform the Subtraction: Subtract $1$ from $-6$: $-6 - 1 = -7$.
7. State the Result: Therefore, $g(-2.3) = -7$.

This step-by-step approach ensures that we don't miss any crucial details and arrive at the correct answer. This methodical approach can be applied to various mathematical problems, helping to avoid errors and build confidence in your problem-solving abilities. Remember, practice makes perfect, so try applying this method to other functions involving floor or ceiling functions to solidify your understanding. Keep up the great work, and you'll be solving complex math problems in no time!

Importance of Precision

When dealing with mathematical functions, especially those involving floor, ceiling, or other special functions, precision is paramount. A small error in applying the definition of a function can lead to a completely incorrect result. In the case of $g(x) = 2\lfloor x \rfloor - 1$, if we mistakenly evaluate $\lfloor -2.3 \rfloor$ as $-2$ instead of $-3$, our final answer would be different. Let's see what happens: If $\lfloor -2.3 \rfloor = -2$, then $g(-2.3) = 2 \cdot (-2) - 1 = -4 - 1 = -5$. This is different from the correct answer of $-7$. This simple example highlights the importance of paying close attention to the details and understanding the precise definitions of mathematical functions. Precision is not just about getting the right answer; it's also about developing a rigorous and disciplined approach to problem-solving. In many real-world applications, mathematical models are used to make critical decisions, and even small errors can have significant consequences. For example, in engineering, an incorrect calculation could lead to a structural failure. In finance, a miscalculation could result in a significant financial loss. Therefore, cultivating a habit of precision and accuracy is essential for success in mathematics and in many other fields. Always double-check your work, pay attention to the details, and strive for a thorough understanding of the underlying concepts. These habits will serve you well throughout your academic and professional life.

Conclusion

In conclusion, evaluating $g(x) = 2\lfloor x \rfloor - 1$ at $x = -2.3$ requires a clear understanding of the floor function and careful application of its definition. By correctly identifying that $\lfloor -2.3 \rfloor = -3$, we were able to substitute this value into the function and obtain the correct result: $g(-2.3) = -7$. This exercise demonstrates the importance of precision in mathematics and the need to pay close attention to the details of each step. Understanding fundamental concepts like the floor function is crucial for success in more advanced mathematical topics. Whether you're studying calculus, analysis, or any other area of mathematics, a solid foundation in these basics will enable you to tackle more complex problems with confidence. So, keep practicing, keep asking questions, and never stop exploring the fascinating world of mathematics! Remember guys, math is like a puzzle, and every piece you solve brings you closer to a complete understanding. And who doesn't love solving puzzles? Keep up the great work, and you'll be a math master in no time! I hope you found this article helpful and informative. Good luck with your mathematical journey!