Calculating F(2) For F(x) = X² - 6x + 8: A Step-by-Step Guide

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Hey guys! Let's dive into this math problem together. We're going to figure out the value of the function f(x) = x² - 6x + 8 when x is equal to 2. It might sound a bit complicated, but trust me, it's easier than it looks. We'll break it down step by step so you can follow along. So, grab your pencils and let's get started!

Understanding the Function

Before we jump into calculations, let’s make sure we understand what the function f(x) = x² - 6x + 8 actually means. In simple terms, a function is like a machine: you put a number in (in this case, our 'x' value), and the machine spits out another number based on the rules defined by the function. Our function here is a quadratic function, which you can recognize because of the x² term. Quadratic functions are super common in math and have a lot of cool properties, but for now, we just need to know how to plug in a value and get an answer.

Breaking Down the Equation

Our equation has three main parts: x², -6x, and +8. Each part plays a role in determining the final output. The x² part means we'll take whatever value we have for x and multiply it by itself. The -6x part means we'll multiply our x value by -6. And the +8 is just a constant, meaning it doesn't change no matter what x is. When we put it all together, we're following the order of operations (PEMDAS/BODMAS), which tells us to handle exponents first (the x²), then multiplication (the -6x), and finally addition and subtraction.

Why This Matters

Understanding the function is crucial because it tells us exactly what to do with our input value (which is 2 in this case). Without understanding the function, we'd just be plugging numbers in randomly, hoping for the best. But with a solid understanding, we can confidently calculate the correct answer. Plus, knowing how functions work is a fundamental concept in algebra and calculus, so it's a skill you'll use again and again. So, let's get ready to plug in that 2 and see what the function spits out!

Calculating f(2)

Okay, now for the fun part: the actual calculation! We know that we need to find f(2), which means we're going to substitute every 'x' in our function with the number 2. So, wherever you see an 'x', imagine replacing it with a '2'. This is the core of evaluating a function at a specific point. We're essentially asking, "What happens when x is 2?" And the function will tell us!

Step-by-Step Substitution

Let's take our function f(x) = x² - 6x + 8 and replace each 'x' with '2'. This gives us: f(2) = (2)² - 6(2) + 8. Notice how we've carefully put the 2 in parentheses to make it clear that we're substituting and not just writing the numbers next to each other. This is especially important when dealing with negative numbers or more complex expressions. Now that we've substituted, we can move on to simplifying the expression.

Following the Order of Operations

Remember PEMDAS/BODMAS? We need to follow the order of operations to get the correct answer. First up are exponents. We have (2)², which means 2 multiplied by itself. 2 * 2 = 4. So, we can replace (2)² with 4. Our equation now looks like this: f(2) = 4 - 6(2) + 8. Next, we handle multiplication. We have -6(2), which means -6 multiplied by 2. -6 * 2 = -12. Now our equation is: f(2) = 4 - 12 + 8. Finally, we do addition and subtraction from left to right. 4 - 12 = -8. So, we have: f(2) = -8 + 8. And -8 + 8 = 0. So, drumroll please… f(2) = 0!

Double-Checking Our Work

It's always a good idea to double-check our work, especially in math. We can quickly run through the steps again to make sure we didn't make any silly mistakes. Substitution: f(2) = (2)² - 6(2) + 8. Exponents: f(2) = 4 - 6(2) + 8. Multiplication: f(2) = 4 - 12 + 8. Addition and Subtraction: f(2) = -8 + 8 = 0. Yep, it still checks out! We're confident that our answer is correct.

Choosing the Correct Alternative

Now that we've calculated f(2) and found it to be 0, we need to look at the multiple-choice options and choose the one that matches our answer. This is a crucial step because even if you do all the math right, you could still get the question wrong if you pick the wrong option. So, let's take a look at the choices:

A) 2 B) 4 C) 6 D) 8

Matching Our Answer

We calculated that f(2) = 0. Looking at the options, none of them directly match our answer. It seems there might be a mistake in the provided alternatives, as 0 is not listed as an option. In a real test scenario, this could indicate a potential error in the question itself. However, it's important to always double-check our work before assuming there's an error. Since we've already verified our calculations and are confident in our result, we can conclude that the correct answer, 0, is not among the given choices.

What to Do in a Real Test

If you encounter a situation like this in a real test, where your calculated answer doesn't match any of the options, here's what you should do: First, quickly review your work one more time to ensure you haven't made any mistakes. If you're still confident in your answer, you might consider selecting the option that's closest to your result, but make a note of the question to revisit if you have time. In some cases, it might be appropriate to bring the discrepancy to the attention of the test proctor or instructor, as there could be an error in the question itself.

Justifying the Answer with Calculation Steps

Justifying your answer is super important, especially in math. It's not enough to just get the right number; you need to show how you got there. This not only helps you get full credit on tests and assignments but also helps you understand the problem better and learn from it. Think of it as telling a story: you're explaining the journey you took to arrive at your destination.

Step-by-Step Explanation

To justify our answer for f(2), we'll walk through each step we took, explaining what we did and why. This makes our reasoning clear and easy to follow. Here's how we can break it down:

  1. Substitution: "First, we substituted x = 2 into the function f(x) = x² - 6x + 8, which gave us f(2) = (2)² - 6(2) + 8."
  2. Exponents: "Next, we evaluated the exponent: (2)² equals 2 multiplied by 2, which is 4. So, our equation became f(2) = 4 - 6(2) + 8."
  3. Multiplication: "Then, we performed the multiplication: -6 multiplied by 2 is -12. Our equation now looked like this: f(2) = 4 - 12 + 8."
  4. Addition and Subtraction: "Finally, we did the addition and subtraction from left to right. 4 minus 12 is -8, so we had f(2) = -8 + 8. -8 plus 8 equals 0."
  5. Conclusion: "Therefore, the value of the function f(x) = x² - 6x + 8 when x = 2 is 0."

Why This Matters

Notice how we didn't just write down the steps; we explained why we did each step. This is key to a good justification. It shows that you understand the underlying concepts and aren't just blindly following rules. Justifying your answer also helps you catch mistakes. If you can't explain a step, it might be a sign that you've made an error. So, always take the time to justify your work – it's a valuable skill in math and beyond.

Conclusion

So, there you have it! We've successfully calculated f(2) for the function f(x) = x² - 6x + 8 and found that it equals 0. We walked through each step, from understanding the function to substituting the value, following the order of operations, and justifying our answer. Even though the correct option wasn't provided in the multiple choices, we were able to confidently determine the correct answer by understanding the process. Remember, math isn't just about getting the right answer; it's about understanding why the answer is correct. Keep practicing, and you'll become a function-calculating pro in no time! And remember guys, always double check your work. You never know when a little mistake can throw off your entire calculation. Keep up the great work, and I'll see you in the next math adventure!