Finding F(3): Solving The Functional Equation

Hey everyone! Let's dive into this interesting math problem where we need to find the value of f(3) given a functional equation. It looks a bit tricky at first, but don't worry, we'll break it down step by step and solve it together. So, grab your thinking caps, and let's get started!

Understanding the Problem

The problem presents us with an equation that relates x and f(x):

x = rac{3f(x) + 2x}{1 + 2f(x)}

Our mission, should we choose to accept it (and we do!), is to figure out what f(3) equals. This means we need to manipulate the equation, substitute x with 3, and then solve for f(3). Sounds like a plan, right? Let's jump into the solution!

Step-by-Step Solution

1. Clearing the Fraction

The first thing we're going to do is get rid of that fraction. Nobody likes dealing with fractions if they can avoid it! To do this, we'll multiply both sides of the equation by the denominator, which is 1 + 2f(x):

$$x(1 + 2f(x)) = 3f(x) + 2x$$

2. Expanding the Left Side

Next, we'll expand the left side of the equation by distributing the x:

$$x + 2xf(x) = 3f(x) + 2x$$

3. Rearranging the Equation

Now, let's rearrange the equation so that all the terms involving f(x) are on one side and the other terms are on the other side. This will help us isolate f(x):

$$2xf(x) - 3f(x) = 2x - x$$

4. Simplifying

Let's simplify both sides of the equation:

$$2xf(x) - 3f(x) = x$$

5. Factoring out f(x)

We can factor out f(x) from the left side of the equation. This is a crucial step to isolate f(x):

$$f(x)(2x - 3) = x$$

6. Isolating f(x)

To finally get f(x) by itself, we'll divide both sides of the equation by (2x - 3):

f(x) = rac{x}{2x - 3}

7. Substituting x = 3

Now comes the moment we've been waiting for! We'll substitute x with 3 to find f(3):

f(3) = rac{3}{2(3) - 3}

8. Calculating f(3)

Let's simplify the expression to get our final answer:

f(3) = rac{3}{6 - 3}

f(3) = rac{3}{3}

$$f(3) = 1$$

So, there we have it! The value of f(3) is 1.

Review and Key Takeaways

Let's quickly recap what we did to solve this problem:

1. Cleared the fraction by multiplying both sides by the denominator.
2. Expanded the equation.
3. Rearranged the equation to group terms involving f(x).
4. Simplified both sides.
5. Factored out f(x).
6. Isolated f(x) by dividing.
7. Substituted x = 3.
8. Calculated f(3).

The key takeaway here is that solving functional equations often involves algebraic manipulation to isolate the function we're interested in. Don't be afraid to rearrange, factor, and simplify!

Why This Matters

You might be wondering, "Why are we even doing this? When will I ever use this in real life?" Well, functional equations pop up in various areas of mathematics and its applications. They help us model relationships between quantities and understand how functions behave. Plus, the problem-solving skills you develop tackling these problems are valuable in many fields, from computer science to engineering.

Practice Makes Perfect

If you found this problem interesting, try tackling similar ones! The more you practice, the more comfortable you'll become with these types of questions. You can look for functional equation problems in math textbooks or online resources. Don't hesitate to try different approaches and see what works best for you.

Exploring Further

If you're curious to delve deeper into functional equations, there are some fascinating topics to explore:

• Types of Functional Equations: There are different types, like Cauchy's functional equation and others, each with its own set of techniques.
• Applications: Learn how functional equations are used in calculus, differential equations, and other areas of math.
• Advanced Techniques: Discover more advanced methods for solving complex functional equations.

Common Mistakes to Avoid

When solving functional equations, it's easy to make a few common mistakes. Here are some things to watch out for:

• Incorrect Algebraic Manipulation: Make sure you're applying algebraic rules correctly when rearranging and simplifying equations.
• Forgetting to Factor: Factoring out the function is often a crucial step, so don't overlook it.
• Not Substituting Correctly: Double-check your substitutions to avoid errors.
• Giving Up Too Soon: Functional equations can be challenging, so don't get discouraged if you don't see the solution right away. Keep trying!

Alternative Approaches

While we solved this problem using algebraic manipulation, there might be other ways to approach it. Sometimes, thinking about the properties of functions or using specific substitutions can lead to a quicker solution. It's always good to explore different methods and see what works best for you.

Conclusion

Great job, guys! We've successfully solved for f(3) in this functional equation. Remember, the key is to break down the problem into smaller steps, use algebraic techniques, and practice regularly. Keep exploring the world of math, and you'll be amazed at what you can discover. If you have any questions or want to discuss this further, feel free to leave a comment below. Happy problem-solving! You've got this!

This problem demonstrates a classic approach to solving functional equations: manipulate the equation to isolate the function and then substitute the given value. By following these steps carefully, we can arrive at the correct answer. Keep practicing, and you'll become a functional equation whiz in no time!