Finding F(4): A Step-by-Step Guide

by Alex Johnson 35 views

Have you ever been faced with a mathematical function and asked to find its value at a specific point? It might seem daunting at first, but with a clear understanding of the function and a few simple steps, you can easily solve it. In this guide, we will walk through finding the value of f(4) for the function f(x) = (1/3) * 4^x. This is a classic example of evaluating a function, and mastering this process will help you tackle more complex mathematical problems.

Understanding the Function

Before we dive into the calculation, let's make sure we understand the function we're working with: f(x) = (1/3) * 4^x. This function tells us that for any input x, we need to perform two operations:

  1. Raise 4 to the power of x (4^x).
  2. Multiply the result by 1/3.

This might seem straightforward, but it's crucial to understand the order of operations. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? It helps us remember the correct order to perform calculations. In our case, we'll first handle the exponentiation (4 to the power of x) and then the multiplication by 1/3.

Now that we've dissected the function, let's move on to the main task: finding f(4).

Step-by-Step Calculation of f(4)

To find f(4), we simply need to substitute x with 4 in the function f(x) = (1/3) * 4^x. This means we'll replace every instance of x in the equation with the number 4. Here’s how it looks:

f(4) = (1/3) * 4^4

Now, let's break down the calculation step by step:

1. Calculate 4^4

The first step is to calculate 4 raised to the power of 4. This means multiplying 4 by itself four times:

4^4 = 4 * 4 * 4 * 4

Let's do the multiplication:

4 * 4 = 16 16 * 4 = 64 64 * 4 = 256

So, 4^4 = 256. This is a crucial step, as getting the exponent calculation wrong will throw off the entire result. Take your time and double-check your calculations to ensure accuracy. Understanding exponents is fundamental to many mathematical concepts, so make sure you're comfortable with this step.

2. Multiply by 1/3

Now that we have 4^4 = 256, we need to multiply this result by 1/3:

(1/3) * 256 = 256/3

This is a simple multiplication of a fraction by a whole number. You can think of it as dividing 256 into three equal parts. The result is 256/3, which is an improper fraction. This means the numerator (256) is larger than the denominator (3).

3. The Result

Therefore, f(4) = 256/3. This is the value of the function f(x) = (1/3) * 4^x when x is equal to 4. We have successfully found f(4) by following a clear, step-by-step approach. This methodical approach is key to solving many mathematical problems.

Understanding the Answer

The answer, 256/3, might seem like just a number, but it represents a specific point on the graph of the function f(x) = (1/3) * 4^x. If you were to graph this function, the point where x = 4 would have a y-coordinate of 256/3. This visual representation can help you understand the behavior of the function and how it changes as x changes.

It's also important to recognize that 256/3 is an exact answer. While you could convert it to a decimal (approximately 85.33), the fraction form is often preferred in mathematics because it preserves the precise value. Decimal approximations can introduce rounding errors, especially if the decimal representation is non-terminating.

Common Mistakes to Avoid

When evaluating functions, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. Here are some of the most frequent errors:

  • Incorrect Order of Operations: As mentioned earlier, the order of operations (PEMDAS) is crucial. Make sure you perform the exponentiation before the multiplication. A common mistake is to multiply 1/3 by 4 first, which leads to an incorrect result. Always prioritize exponents before multiplication and division.
  • Miscalculating the Exponent: Exponents can be tricky, especially when dealing with larger numbers. Double-check your calculations to ensure you're multiplying the base (in this case, 4) by itself the correct number of times. Using a calculator can be helpful for larger exponents, but make sure you understand the concept behind the calculation.
  • Forgetting the Fraction: Don't forget to multiply the result of the exponentiation by the fraction (1/3). It's easy to get caught up in the exponent calculation and forget the final step. Always double-check that you've performed all the operations in the function.
  • Rounding Errors: If you convert the fraction to a decimal, be mindful of rounding errors. If the decimal representation is non-terminating, rounding too early can lead to inaccuracies. It's generally best to keep the answer in fraction form unless a decimal approximation is specifically requested.

By being aware of these common mistakes and carefully checking your work, you can increase your chances of getting the correct answer.

Practice Makes Perfect

The best way to master evaluating functions is to practice! Try working through similar problems with different functions and different values of x. The more you practice, the more comfortable you'll become with the process. Consistent practice is the key to success in mathematics.

Here are a few practice problems you can try:

  1. Find f(2) for f(x) = 2 * 3^x
  2. Find g(5) for g(x) = (1/2) * 5^x
  3. Find h(3) for h(x) = (2/5) * 2^x

Working through these problems will help solidify your understanding of evaluating functions. Remember to follow the step-by-step approach we discussed and double-check your calculations.

Conclusion

Finding f(4) for the function f(x) = (1/3) * 4^x is a great example of how to evaluate a function at a specific point. By understanding the function, following the correct order of operations, and avoiding common mistakes, you can confidently solve this type of problem. Remember, practice is key to mastering mathematical concepts, so keep working at it! We've broken down the problem into easy-to-understand steps:

  1. We started by understanding the function and what it represents.
  2. We then substituted x with 4 in the function.
  3. We calculated 4^4.
  4. We multiplied the result by 1/3.
  5. Finally, we understood the significance of the answer.

By following these steps, you can tackle similar problems with confidence. Remember to always double-check your work and practice regularly to improve your skills. Exploring additional resources can further enhance your understanding of functions and their applications. Check out Khan Academy's Functions and equations for more learning materials.