Calculating H(3) - 2h(1) A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem that involves function evaluation. We're given a function, h(x) = 2x + 1, and our mission is to calculate the value of h(3) - 2h(1). This might seem a bit daunting at first, but trust me, it's super straightforward once you break it down. We'll go through each step together, so you can ace similar problems in the future. So grab your pencils and let's get started!

Understanding the Function h(x)

Before we jump into the calculation, let's make sure we're all on the same page about what the function h(x) actually means. The function h(x) = 2x + 1 is a linear function. In simple terms, it's a rule that tells us what to do with any input value we give it. Think of it like a machine: you feed it a number (x), it performs some operations on it (in this case, multiplying by 2 and adding 1), and then it spits out a new number (the result of h(x)). The 'x' is what we call the independent variable, and it represents the input we're giving to the function. The 'h(x)' is the dependent variable, which is the output we get after applying the function's rule. So, if we input 3 into the function, we write it as h(3), which means we're plugging in 3 for 'x' in the expression 2x + 1. This concept of input and output is fundamental to understanding functions, and it's crucial for solving problems like the one we have today. To really nail this down, try thinking of other functions you might have encountered. For example, f(x) = x^2 is another function that squares whatever input you give it. The key is recognizing that the function is just a set of instructions that operates on the input variable. Once you've got that down, evaluating functions becomes a breeze!

Step-by-Step Calculation of h(3)

Okay, now that we've got a solid understanding of what h(x) means, let's tackle the first part of our problem: calculating h(3). Remember, h(3) means we're substituting 'x' with 3 in our function h(x) = 2x + 1. So, wherever we see an 'x', we're going to replace it with a 3. This gives us h(3) = 2(3) + 1. See how we've simply swapped the 'x' with the value 3? The parentheses around the 3 indicate multiplication, which is super important to remember in mathematical expressions. Now, we just need to follow the order of operations (PEMDAS/BODMAS) to simplify this expression. First up is multiplication: 2 multiplied by 3 equals 6. So, our equation now looks like this: h(3) = 6 + 1. The only operation left is addition, and 6 plus 1 is, of course, 7. Therefore, we've successfully calculated that h(3) = 7. Woohoo! We've conquered the first hurdle. It's really as simple as plugging in the value and simplifying. Don't be intimidated by the function notation; just think of it as a set of instructions. This step is crucial because it forms the foundation for the rest of the problem. We'll use this value in our final calculation, so make sure you've got it locked in. Now, let's move on to the next part – finding h(1) – and you'll see how smoothly this process goes when you understand the basics.

Step-by-Step Calculation of h(1)

Alright, let's keep the momentum going! Now we need to calculate h(1). Just like before, this means we're going to substitute 'x' with 1 in our function h(x) = 2x + 1. Following the same process as before, we replace 'x' with 1, which gives us h(1) = 2(1) + 1. Again, the parentheses remind us that we're multiplying 2 by 1. This is a super important detail to pay attention to, as it can change the outcome if you miss it. Now, let's simplify. First, we perform the multiplication: 2 multiplied by 1 is simply 2. So our equation now looks like this: h(1) = 2 + 1. We're almost there! The final step is addition: 2 plus 1 equals 3. Therefore, we've found that h(1) = 3. Great job! We've successfully calculated the value of h(1). Notice how the process is exactly the same as when we calculated h(3)? That's the beauty of functions; once you understand the rule, you can apply it to any input. Now that we have both h(3) and h(1), we're in the home stretch. We're ready to put these values together to find the final answer. Remember, the key here is consistency and attention to detail. Keep practicing these substitutions, and you'll become a pro in no time!

Calculating 2h(1)

Before we can calculate the final answer, h(3) - 2h(1), there's one small step we need to take: calculating 2h(1). We already know that h(1) = 3, thanks to our previous calculation. So, 2h(1) simply means 2 multiplied by the value of h(1), which is 3. This is a classic example of how mathematical notation can be super concise. The absence of any symbol between the 2 and the h(1) implies multiplication. Therefore, we have 2h(1) = 2 * 3. This is a straightforward multiplication problem. 2 multiplied by 3 equals 6. So, 2h(1) = 6. Excellent! We've now calculated the value of 2h(1), which is a crucial component of our final calculation. This step highlights the importance of breaking down complex problems into smaller, manageable parts. By calculating h(1) first and then multiplying it by 2, we avoided making a potential mistake by trying to do everything at once. Keep this strategy in mind as you tackle other math problems; breaking things down often makes the solution much clearer. We're now just one step away from the grand finale, so let's bring it home!

Final Calculation: h(3) - 2h(1)

Okay, guys, the moment we've been working towards is finally here! We're ready to calculate the final answer: h(3) - 2h(1). Remember, we've already done all the hard work. We know that h(3) = 7 (from our first calculation) and 2h(1) = 6 (from our previous step). Now, we simply need to substitute these values into the expression. So, h(3) - 2h(1) becomes 7 - 6. This is a simple subtraction problem. 7 minus 6 equals 1. Therefore, our final answer is 1! Hooray! We've successfully calculated h(3) - 2h(1) given the function h(x) = 2x + 1. Give yourselves a pat on the back; you've navigated through the entire problem step-by-step. This final calculation demonstrates how all the individual pieces we calculated earlier fit together to give us the solution. It's like building a puzzle; each piece is important, and only when they're all in place do you see the complete picture. Remember, the key to success in math is to break down complex problems into smaller, more manageable steps, just like we did here. Now, let's recap everything we've learned to solidify our understanding.

Recap and Conclusion

Let's take a moment to recap what we've accomplished today. We started with the function h(x) = 2x + 1 and the task of calculating h(3) - 2h(1). We broke this problem down into smaller, more manageable steps: First, we understood the function notation and what it means to evaluate h(x) for different values of x. Then, we calculated h(3) by substituting x with 3 in the function, giving us h(3) = 7. Next, we calculated h(1) by substituting x with 1, which resulted in h(1) = 3. We then calculated 2h(1) by multiplying our value for h(1) by 2, giving us 2h(1) = 6. Finally, we performed the subtraction h(3) - 2h(1), substituting the values we calculated to get 7 - 6 = 1. Therefore, our final answer is h(3) - 2h(1) = 1. Awesome job! You've successfully navigated through this function evaluation problem. This exercise demonstrates the importance of breaking down complex problems into simpler steps, understanding function notation, and applying the order of operations. Remember, practice makes perfect. The more you work with functions and similar problems, the more comfortable and confident you'll become. So, keep practicing, and you'll be a math whiz in no time! Thanks for joining me on this mathematical adventure. Keep exploring, keep learning, and most importantly, keep having fun with math!