Calculating Function Values: F(x) = 3^(x-2) For X=4 & X=6

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Calculating Function Values for f(x) = 3^(x-2)

Hey guys! Today, we're diving into the world of functions, specifically focusing on calculating function values. We've got a cool function to work with: f(x) = 3^(x-2). Our mission, should we choose to accept it (and we do!), is to figure out what the function spits out when we plug in x = 4 and x = 6. Don't worry, it's not as intimidating as it sounds! Let's break it down step by step. Understanding how to evaluate functions is a foundational skill in mathematics, and it opens doors to more advanced concepts like calculus and mathematical modeling. So, grab your thinking caps, and let's get started!

f(4): Let's Plug in x = 4

First, let's tackle f(4). What this notation means is that we're going to replace every instance of 'x' in our function with the number 4. So, wherever we see 'x' in the equation f(x) = 3^(x-2), we'll swap it out for a 4. This is the core concept of evaluating a function at a specific point. We are essentially asking, "What is the output of the function when the input is 4?" It’s like a mathematical machine – you feed it an input (in this case, 4), and it processes it according to the function's rule (3 raised to the power of (x-2)), and then it gives you an output.

So, our equation transforms like this:

  • f(4) = 3^(4-2)

Now, we simplify the exponent first. Remember the order of operations (PEMDAS/BODMAS)? Parentheses/Brackets come before Exponents/Orders. Inside the parentheses, we have 4 - 2, which equals 2. This is a crucial step. If we were to calculate the exponent incorrectly, the whole answer would be off! So, let's make sure we have this right: 4 minus 2 indeed equals 2. We are on the right track!

Our equation now looks even simpler:

  • f(4) = 3^2

What does 3^2 mean? It means 3 multiplied by itself. Think of it as 3 to the power of 2, or 3 squared. This is a fundamental concept in exponents. When a number is raised to a power, it means you multiply the number by itself that many times. So, 3 squared means 3 times 3.

Therefore:

  • f(4) = 3 * 3 = 9

And there we have it! The value of the function f(x) when x = 4 is 9. We have successfully evaluated the function at the point x = 4. This result tells us a specific point on the graph of this function: (4, 9). When x is 4, y (or f(x)) is 9. This is a clear and concise answer, demonstrating our understanding of function evaluation. Great job!

f(6): Let's Plug in x = 6

Alright, now let's move on to the second part of our quest: calculating f(6). Just like before, this means we're going to replace every 'x' in the function f(x) = 3^(x-2) with the number 6. We're essentially repeating the process we just learned, but with a different input value. This is great practice, because the more we work with these concepts, the better we understand them. Remember, practice makes perfect, especially in mathematics! This repetition will help solidify the process in our minds.

So, let's substitute x = 6 into our function:

  • f(6) = 3^(6-2)

Again, we need to simplify the exponent first, following the order of operations. Inside the parentheses, we have 6 - 2. What does that equal? Take a moment to calculate it. This is another crucial step, just like before. An incorrect calculation here will throw off the entire result. So, we want to be careful and make sure we get it right. We want to ensure a solid foundation for the next steps.

6 minus 2 equals 4. Excellent! Now our equation looks like this:

  • f(6) = 3^4

Now, what does 3^4 mean? It means 3 multiplied by itself four times. It's 3 * 3 * 3 * 3. This is another example of a number raised to a power. The power indicates how many times the base number (in this case, 3) is multiplied by itself. So, 3 to the power of 4 means we have four 3s multiplied together.

Let's break this down step by step to make it easier to calculate. We can first calculate 3 * 3, which is 9. Then we have 9 * 3, which is 27. And finally, 27 * 3...

  • f(6) = 3 * 3 * 3 * 3 = 81

So, the value of the function f(x) when x = 6 is 81. We have successfully evaluated the function at x = 6! This tells us another point on the graph of the function: (6, 81). When the input is 6, the output is a whopping 81! We did it! We’ve calculated the function’s output for another input value, further solidifying our understanding of function evaluation. Fantastic work, guys!

In Conclusion: The Function Values

So, to recap, we've successfully calculated the values of the function f(x) = 3^(x-2) for x = 4 and x = 6. We found that:

  • f(4) = 9
  • f(6) = 81

These results tell us specific points on the graph of the function. Understanding how to calculate function values is a fundamental skill in mathematics. It's like learning the alphabet before you can write a sentence. Function evaluation is a building block for more complex concepts. Now you can confidently plug in values for x and determine the corresponding output of this function. This skill is not only useful for this specific function but also for any function you encounter in the future. It’s a versatile tool in your mathematical toolbox.

This exercise also highlights the importance of following the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. We saw how simplifying the exponent first was crucial in both cases. Skipping this step or performing the operations in the wrong order would have led to incorrect answers. The order of operations is a consistent rule that applies across all mathematical calculations. Mastering it is essential for success in math.

Keep practicing, and you'll become function-evaluation pros in no time! Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, don’t be afraid to tackle more problems, explore different functions, and challenge yourself. The world of mathematics is vast and exciting, and the more you learn, the more you’ll appreciate its beauty and power. So, keep up the great work! You are on your way to becoming mathematical masters!