Calculating G(x): A Step-by-Step Guide

Hey everyone! Let's dive into how to find the values of a function, specifically $g(x)$, given its definition. In this case, $g(x)$ is a piecewise function, meaning it has different rules depending on the value of x. Understanding piecewise functions is super important in math, so let's break it down, step by step, so that you can easily grasp the concept. We will calculate the values of $g(2.5)$, $g(3)$, and $g(3.5)$ based on the provided definition of $g(x)$. This will show you exactly how to navigate this type of problem, making it a breeze for you!

Understanding the Piecewise Function $g(x)$

Before we start, let's take a look at the definition of our function, $g(x)$. The function is defined as follows:

$g(x)=\begin{cases} 1003.00 & \text{ if } 0 \leq x < 1 \\ 1060.00 & \text{ if } 1 \leq x < 2 \\ 1124.90 & \text{ if } 2 \leq x < 3 \\ 1194.00 & \text{ if } 3 \leq x < 4 \\ 1263.40 & \text{ if } 4 \leq x < 5 \end{cases}$

This means that the value of $g(x)$ depends on the interval in which x falls. For example, if x is between 0 and 1 (including 0 but not including 1), then $g(x)$ is 1003.00. If x is between 1 and 2 (including 1 but not including 2), then $g(x)$ is 1060.00, and so on. Now, let's find the values for $g(2.5)$, $g(3)$, and $g(3.5)$. This is how the magic happens, guys! The key is to match the input value (x) to the correct interval and then use the corresponding output value. It's like a matching game! You look at the input, find its home in the function's definition, and then you know its output. Remember, in a piecewise function, each interval has a specific rule associated with it. Now, let's get into the specifics of finding those values.

To really get this, remember that the function is a set of rules. Each rule applies only to a specific range of x values. So when we want to find $g(2.5)$, we're asking: "Which rule applies when x is 2.5?" Same goes for $g(3)$ and $g(3.5)$. The crucial thing is always figuring out which range your x value fits into. That range tells you which piece of the function you're using. So, don't sweat it; we will break down each calculation.

Step-by-Step Guide to Calculating g(x)

Let's get down to the actual calculation. Here's a breakdown for each value: $g(2.5)$, $g(3)$, and $g(3.5)$. We'll walk through each one so you see exactly how to approach it. The core principle stays the same: locate where your x value falls within the intervals defined in the function, then grab the corresponding value of $g(x)$. It's a simple process if you understand the ranges. Let's make it easy, shall we? Ready? Let's go!

Calculating $g(2.5)$

To find $g(2.5)$, we need to determine which interval 2.5 belongs to. Looking at the function definition, we can see that:

• $1 \leq x < 2$ corresponds to $g(x) = 1060.00$
• $2 \leq x < 3$ corresponds to $g(x) = 1124.90$

Since 2.5 falls within the interval $2 \leq x < 3$, the value of $g(2.5)$ is 1124.90. This means when x is 2.5, $g(x)$ is 1124.90. Nice and easy, right? You just need to match the x to the correct interval, and bam! You've got your answer. Remember, the value of x in the given function determines which part of the piecewise function you're going to use.

So, think of the function's definition as a set of rules. Each rule applies only under specific conditions. In this case, our condition is the value of x. The different ranges define which rule to use. In this instance, when $x = 2.5$, the rule that applies tells us that $g(x)$ is 1124.90. That's the beauty of piecewise functions! The value depends directly on the interval of the input variable.

Calculating $g(3)$

Next, let's find $g(3)$. We need to see which interval includes the number 3. Examining the function definition, we find that:

• $2 \leq x < 3$ corresponds to $g(x) = 1124.90$
• $3 \leq x < 4$ corresponds to $g(x) = 1194.00$

Notice that the interval $3 \leq x < 4$ includes the number 3 (because it says x is greater than or equal to 3). Therefore, $g(3) = 1194.00$. So when x is exactly 3, the value of $g(x)$ is 1194.00. Note the importance of the symbols $\leq$ and $<$. The function explicitly defines these ranges. Always pay close attention to the less-than and greater-than signs when dealing with piecewise functions, as these determine the values for which each part of the function applies. If there were any other number, it would be another value, but since it is 3, then it is 1194.00. Remember this when you are working with these types of functions.

So here is the key takeaway: for $x = 3$, we apply the rule $g(x) = 1194.00$. It's a direct mapping. You plug in your x value, check the intervals, and take the corresponding output. It's really that simple! Don't overthink it, trust the definition of the function, and it will guide you to the correct answer. The critical thing here is understanding how each part of the piecewise function works independently based on the interval. When you see a problem like this, remember to look at the definitions, and apply the correct values.

Calculating $g(3.5)$

Finally, let's calculate $g(3.5)$. We need to identify the correct interval for 3.5. Going back to the function definition:

• $3 \leq x < 4$ corresponds to $g(x) = 1194.00$
• $4 \leq x < 5$ corresponds to $g(x) = 1263.40$

Since 3.5 falls within the interval $3 \leq x < 4$, the value of $g(3.5)$ is 1194.00. This means for x = 3.5, $g(x)$ = 1194.00. Spot on! You see, all that's needed is to match the value of x to the correct definition. Piecewise functions are not as complicated as they might seem initially. The trick is to understand which rule applies to your specific x value. The function provides all the information you need in the intervals. You've now mastered the skill of finding the values for piecewise functions.

So, to recap: $g(2.5) = 1124.90$, $g(3) = 1194.00$, and $g(3.5) = 1194.00$. You did it, guys! This process is straightforward: Identify the interval your x value falls in and grab the corresponding value of $g(x)$.

Conclusion: Mastering Piecewise Functions

Alright, folks, we've walked through the calculations for $g(2.5)$, $g(3)$, and $g(3.5)$, using a piecewise function. The key takeaway? Understand the intervals and match your input (x) to the correct rule. It's a fundamental concept in mathematics. Remember, piecewise functions break down a function into different rules based on the range of x values. So, it's really like solving several mini-problems, each with its own rule! The key is always to identify the interval that your x value falls into and then apply the rule associated with that interval. Remember to check whether the input value is within the closed or open interval.

By following these steps, you can confidently evaluate any piecewise function. Keep practicing, and you'll become a pro in no time! Piecewise functions might seem daunting at first, but with a bit of practice, you can get the hang of them. Just remember to carefully examine the intervals. So the next time you encounter a piecewise function, you'll know exactly what to do! Now you're ready to tackle more complex math problems involving piecewise functions. Good luck, and keep practicing!