Finding F(4) Given F(x) = (1/3) * 4^x A Step-by-Step Guide
Hey guys! Let's dive into this math problem where we need to find the value of F(4) using the function F(x) = (1/3) * 4^x. It might seem a bit daunting at first, but trust me, it’s super straightforward once we break it down. We’ll go through each step, making sure you understand exactly how we arrive at the correct answer. So, grab your calculators (or just your thinking caps!) and let’s get started!
Understanding the Function
Before we jump into plugging in numbers, let's make sure we fully grasp what the function F(x) = (1/3) * 4^x means. At its heart, this is an exponential function. The key thing to notice here is that 'x' is in the exponent, which means the value of the function will change dramatically as 'x' changes. Exponential functions are used all over the place in the real world, from calculating compound interest to modeling population growth. So, understanding how they work is pretty crucial.
In our case, the function tells us to take 4 raised to the power of 'x', and then multiply the result by 1/3. That’s it! The (1/3) part is just a constant factor that scales the value of 4^x. Think of it like this: no matter what 'x' is, we're always going to end up dividing our result by 3. This constant factor is important because it shifts the graph of the function up or down (in this case, it compresses the graph vertically).
Now, let's talk about the base, which is 4 in our function. The base is the number that’s being raised to the power of 'x'. The base plays a huge role in how the function behaves. A base greater than 1 (like our 4) means the function will increase as 'x' increases. If the base were between 0 and 1 (say, 1/2), the function would decrease as 'x' increases. Understanding this behavior helps you make quick predictions about the function's values.
So, to recap, the function F(x) = (1/3) * 4^x is an exponential function where we take 4 to the power of 'x', and then multiply by 1/3. Keep this in mind as we move on to the next part, where we actually calculate F(4). Knowing the basics of the function is half the battle, and now we're well-equipped to tackle the problem head-on!
Step-by-Step Calculation of F(4)
Okay, now that we understand the function, let’s get down to business and calculate F(4). This is where the fun really begins because we get to put our knowledge into action. Remember, the goal here is to find the value of the function when x is equal to 4. So, we're essentially substituting 4 for every 'x' we see in the function. Ready? Let's do it!
The first thing we need to do is replace 'x' with 4 in our function. So, F(x) = (1/3) * 4^x becomes F(4) = (1/3) * 4^4. See how we just swapped 'x' for 4? That’s the main trick here. Now, we just need to simplify the right side of the equation.
Next up, we need to deal with that exponent. We have 4^4, which means 4 raised to the power of 4. This is the same as multiplying 4 by itself four times: 4 * 4 * 4 * 4. Let’s break it down step-by-step to make it super clear.
First, 4 * 4 equals 16. So, we've got 16 * 4 * 4. Then, 16 * 4 gives us 64. Now, we have 64 * 4. Finally, 64 * 4 equals 256. So, 4^4 is 256. Phew! We got there. Exponents can look intimidating, but they're just repeated multiplication when you break them down like this.
Now we can rewrite our equation as F(4) = (1/3) * 256. We’ve taken care of the exponent, and we're one step closer to the final answer. The last step is to multiply 256 by 1/3. Remember, multiplying by 1/3 is the same as dividing by 3. So, we need to calculate 256 / 3.
When we divide 256 by 3, we get 85 with a remainder of 1. But since we're dealing with fractions, we'll express this as an improper fraction. So, 256 divided by 3 is 256/3. This is our final value for F(4). We’ve done it! We’ve successfully calculated the value of the function at x = 4.
So, just to recap, we started with F(x) = (1/3) * 4^x, substituted 4 for 'x' to get F(4) = (1/3) * 4^4, calculated 4^4 as 256, and then multiplied by 1/3 to get 256/3. Each step is logical and builds on the previous one. With practice, these types of calculations become second nature.
Matching the Result with the Options
Alright, we've done the hard work and found that F(4) = 256/3. Great job, guys! Now, the final step is to match our result with the options provided in the problem. This is super important because it’s easy to make a small mistake somewhere along the way, and checking against the options is a good way to catch any errors.
Let's take a look at the options:
A. 8/3 B. 64/3 C. 16/3 D. 256/3
Comparing our calculated result, 256/3, with the options, it's clear that option D, 256/3, is the correct answer. Sometimes, the answer just jumps out at you, and this is one of those times! This is always a satisfying moment because it confirms that we've followed the correct steps and arrived at the right solution.
But what if our answer wasn't directly there? What if we had made a small mistake and gotten a different result? That's where this step becomes even more valuable. If our answer didn't match any of the options, it would be a signal to go back and carefully review our calculations. We might have made a mistake in calculating the exponent, or perhaps we slipped up with the multiplication or division. By double-checking, we can catch these errors and make sure we submit the correct answer.
In a multiple-choice question like this, the options are there to guide you. They provide a set of possible answers, and one of them is definitely the right one. So, always take the time to compare your result with the options and make sure everything lines up. It’s a small step that can make a big difference in your overall score. Plus, it gives you that extra bit of confidence knowing you've nailed it!
Key Takeaways and Tips for Similar Problems
Okay, guys, we've successfully found F(4) using the function F(x) = (1/3) * 4^x and matched our answer with the options. High five! But more than just solving this specific problem, it’s super valuable to take away some key concepts and tips that you can apply to similar problems in the future. So, let's recap the main ideas and talk about how to tackle these types of questions with confidence.
Understanding Functions
The most important takeaway is understanding what a function actually represents. A function is like a machine: you put something in (the input, like 'x'), and it spits something out (the output, like F(x)). In this case, our function F(x) tells us exactly what to do with the input 'x': we raise 4 to the power of 'x', and then multiply the result by 1/3. Grasping this concept is huge because it’s the foundation for all sorts of math problems.
Step-by-Step Calculation
When dealing with exponential functions, breaking the problem down into clear steps is key. First, substitute the value of 'x' into the function. Then, tackle the exponent. Remember that an exponent is just repeated multiplication, so take it one step at a time. Finally, do any remaining multiplication or division to get your final result. This step-by-step approach keeps things organized and reduces the chance of making a mistake.
Checking Your Work
Always, always, always check your work! Especially in a multiple-choice question, comparing your answer with the options is a lifesaver. If your calculated answer doesn't match any of the options, that’s a red flag to go back and review your steps. Did you make a mistake with the exponent? Did you multiply or divide correctly? Catching these errors is part of the problem-solving process, and it’s what turns a good math student into a great one.
Tips for Similar Problems
- Practice makes perfect: The more you work with functions and exponents, the more comfortable you’ll become. Try solving similar problems with different functions and values of 'x'.
- Know your exponent rules: Understanding how exponents work is crucial. Remember that a^b means 'a' multiplied by itself 'b' times. Also, be familiar with other exponent rules, like a^(b+c) = a^b * a^c and (ab)c = a^(b*c).
- Use a calculator wisely: A calculator can be a powerful tool, especially for larger exponents. But make sure you know how to use it correctly. Enter the numbers carefully and double-check your results.
- Stay organized: Keep your work neat and organized. Write down each step clearly so you can easily review your calculations later if needed.
By keeping these takeaways and tips in mind, you’ll be well-equipped to tackle all sorts of problems involving functions and exponents. Math might seem tricky sometimes, but with a solid understanding of the basics and a step-by-step approach, you can conquer any challenge!
Conclusion
So, there you have it! We've successfully navigated through the process of finding F(4) using the function F(x) = (1/3) * 4^x. We started by understanding the function, then we calculated F(4) step-by-step, matched our result with the options, and finally, we recapped the key takeaways and tips for similar problems. What a journey! You guys did an awesome job sticking with it, and I hope you feel a real sense of accomplishment.
Remember, math isn't just about getting the right answer; it's about understanding the process and building your problem-solving skills. Each time you work through a problem like this, you're strengthening your ability to think critically and logically. These are skills that will serve you well in all areas of life, not just in math class.
The key to success in math (and really, in anything) is practice. Don't be afraid to tackle challenging problems, and don't get discouraged if you don't get it right away. Every mistake is a learning opportunity, a chance to deepen your understanding and improve your skills. So, keep practicing, keep asking questions, and keep pushing yourself to learn more.
And remember those key takeaways we talked about: understanding functions, breaking down problems into steps, checking your work, and using the tools at your disposal (like calculators and exponent rules). These are the principles that will guide you through any math challenge, no matter how big or small.
So, next time you encounter a problem like this, remember the steps we followed, the tips we discussed, and the confidence you gained from working through this example. You've got this! Keep up the great work, and I'm excited to see all the math mountains you'll climb!
