Evaluating G(-2.3) For G(x) = 2[x] - 1: A Step-by-Step Guide

Hey guys! Let's dive into a math problem where we need to figure out the value of a function. Specifically, we're looking at the function g(x) = 2[x] - 1, and our mission, should we choose to accept it, is to find out what g(-2.3) is. Sounds like fun, right? Don't worry, it's not as scary as it looks. We'll break it down step by step so it's super easy to follow. This is a classic example that combines function evaluation with the concept of the greatest integer function, also known as the floor function. Understanding this type of problem is super useful for anyone studying functions in algebra or calculus. So, let’s get started and make sure we nail this concept! We'll explore the greatest integer function, the process of substitution, and the arithmetic involved, making sure you're confident in tackling similar problems in the future. Remember, practice makes perfect, so let's get our practice in!

Understanding the Greatest Integer Function

Before we even think about plugging -2.3 into our function, we need to chat about what this [x] thing actually means. This notation represents the greatest integer function, also sometimes called the floor function. Essentially, what it does is take any real number (like -2.3) and rounds it down to the nearest integer. Think of it as finding the largest integer that is less than or equal to your number. This concept is crucial for understanding the behavior of functions that involve it, and it pops up in various areas of mathematics, from number theory to calculus. So, let’s break it down further to really get a grip on it.

What is the Greatest Integer Function?

The greatest integer function, denoted by [x], gives the greatest integer less than or equal to x. It's like finding the floor – you go down to the nearest whole number. This might sound a bit abstract, so let’s look at some examples to make it crystal clear. If we have a positive number like 3.7, the greatest integer less than or equal to 3.7 is 3. Simple enough, right? But what happens when we throw negative numbers into the mix? This is where it can get a little trickier, but stick with me!

Examples to Clarify

Let's take -2.3 as our key example since that's what we'll be working with later. The greatest integer less than or equal to -2.3 is -3. Why? Because -3 is the first integer you hit as you move left on the number line from -2.3. It’s lower than -2.3. Think of it like this: if you're at -2.3 on the number line and you can only move to whole numbers, you have to step back to -3, not forward to -2. Other examples include [5.9] = 5 (since 5 is the greatest integer less than or equal to 5.9) and [-1.5] = -2 (because -2 is the greatest integer less than or equal to -1.5). Getting comfortable with negative numbers and the greatest integer function is key to solving our problem!

Why is This Important?

Understanding the greatest integer function is super important because it's used in various fields like computer science (for algorithms and data structures), engineering (for signal processing), and, of course, mathematics. It helps us model situations where we need to deal with discrete steps or boundaries. Knowing how it works ensures you won't get tripped up when you see this notation. Plus, it's a foundational concept for more advanced topics. So, mastering this now sets you up for success later on!

Evaluating g(-2.3)

Now that we've got the greatest integer function down, let’s get back to our main goal: figuring out g(-2.3) when g(x) = 2[x] - 1. The key here is substitution. We're going to replace the x in our function with the specific value we're interested in, which is -2.3. This is a fundamental skill in algebra and calculus, and it’s something you’ll use all the time. Think of it as plugging in a value into a machine – you put something in, and something else comes out based on the function's rules. In our case, the "machine" is the function g(x), and we're about to see what happens when we "feed" it -2.3.

Step 1: Substitution

So, everywhere we see an x in the function g(x) = 2[x] - 1, we're going to replace it with -2.3. This gives us:

g(-2.3) = 2[-2.3] - 1

See? Not too scary! We’ve just swapped x for -2.3. This sets us up perfectly for the next step, which involves dealing with that greatest integer function we just learned about.

Step 2: Apply the Greatest Integer Function

Remember, [-2.3] means “the greatest integer less than or equal to -2.3”. We already figured out that [-2.3] = -3. So, we can now replace [-2.3] with -3 in our equation. This simplifies things nicely:

g(-2.3) = 2(-3) - 1

Now we're down to some basic arithmetic. We've taken care of the function part, and now it’s all about multiplying and subtracting. Keep those order of operations in mind (PEMDAS/BODMAS) – multiplication comes before subtraction.

Step 3: Arithmetic

Let’s finish this off! First, we perform the multiplication:

2 * (-3) = -6

So our equation now looks like this:

g(-2.3) = -6 - 1

Finally, we do the subtraction:

-6 - 1 = -7

And there we have it! We've found that g(-2.3) = -7. High five! You've successfully evaluated the function for a specific value. This process of substitution and simplification is a cornerstone of function work, so you've added a great tool to your math toolbox. Let’s take a moment to recap what we’ve done.

Final Answer and Summary

Alright, guys, let's wrap this up! We started with the function g(x) = 2[x] - 1 and the challenge of finding g(-2.3). We navigated the ins and outs of the greatest integer function, substituted -2.3 into our function, and performed the necessary arithmetic. And guess what? We nailed it!

The Solution

We found that g(-2.3) = -7. That's the final answer! Pat yourselves on the back – you've conquered this problem.

Quick Recap of the Steps

1. Understand the Greatest Integer Function: We made sure we knew what [x] meant and how it works, especially with negative numbers.
2. Substitution: We plugged -2.3 into the function, replacing x with -2.3.
3. Apply the Greatest Integer Function: We figured out that [-2.3] = -3.
4. Arithmetic: We did the multiplication and subtraction to get our final answer.

Why This Matters

This might seem like a small problem, but it touches on some super important mathematical concepts. You've practiced function evaluation, worked with the greatest integer function, and honed your arithmetic skills. These are the building blocks for more advanced math topics, so you're setting yourself up for success. Plus, you've gained confidence in tackling tricky problems. Remember, math is like a muscle – the more you use it, the stronger it gets!

Keep Practicing!

If you enjoyed this or found it helpful, keep practicing! Try evaluating the function for different values, or explore other functions that involve the greatest integer function. The more you practice, the more comfortable and confident you'll become. And remember, math is awesome! You’ve got this!