Find F(2x) If F(x) = (3x-1)/(x+2): Step-by-Step Solution

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Hey guys! Today, we're diving into a fun little math problem. We've got a function, f(x), and our mission is to figure out what f(2x) looks like. Sounds like a plan? Awesome, let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we all understand what the problem is asking. We are given the function:

f(x)=3x−1x+2f(x) = \frac{3x - 1}{x + 2}

Our goal is to find f(2x). This means we need to replace every x in the original function with 2x. It’s like a substitution game, and we’re about to win it!

Step-by-Step Solution

Step 1: Substitute x with 2x

Alright, first things first, we're going to replace every x in the function f(x) with 2x. This is the heart of the problem, so let’s take it slow and steady to avoid any mistakes. Our original function is:

f(x)=3x−1x+2f(x) = \frac{3x - 1}{x + 2}

Now, replace x with 2x:

f(2x)=3(2x)−1(2x)+2f(2x) = \frac{3(2x) - 1}{(2x) + 2}

Step 2: Simplify the Expression

Okay, now that we've made the substitution, let's simplify the expression. This involves doing the multiplication in the numerator and keeping the denominator as is for now.

f(2x)=6x−12x+2f(2x) = \frac{6x - 1}{2x + 2}

At this point, we have a simplified expression. But, can we simplify it further? Sometimes, you can factor out common terms to make the expression even simpler. In this case, let’s see if we can factor anything out of the numerator or the denominator.

Step 3: Check for Further Simplification

Looking at the numerator, 6x - 1, there's no common factor that we can factor out. The denominator, 2x + 2, however, has a common factor of 2. Let's factor that out:

2x+2=2(x+1)2x + 2 = 2(x + 1)

So, we can rewrite our expression as:

f(2x)=6x−12(x+1)f(2x) = \frac{6x - 1}{2(x + 1)}

Now, we need to check if anything cancels out between the numerator and the denominator. In this case, 6x - 1 and 2(x + 1) don't have any common factors, so we can't simplify further.

Step 4: State the Final Answer

Since we can't simplify any further, we've reached our final answer. The expression for f(2x) is:

f(2x)=6x−12(x+1)f(2x) = \frac{6x - 1}{2(x + 1)}

Or, alternatively:

f(2x)=6x−12x+2f(2x) = \frac{6x - 1}{2x + 2}

Both forms are correct, but the factored form can sometimes be more useful depending on what you need to do with the expression next.

Common Mistakes to Avoid

When working on problems like this, it’s easy to make a few common mistakes. Here are some things to watch out for:

  1. Incorrect Substitution: Make sure you replace every instance of x with 2x. It’s easy to miss one!
  2. Arithmetic Errors: Double-check your arithmetic when simplifying the expression. A small mistake can throw off the entire answer.
  3. Forgetting to Distribute: When simplifying, ensure you distribute correctly. For example, 3(2x) should be 6x, not something else.
  4. Incorrect Factoring: Always double-check your factoring to make sure you've done it correctly. Factoring errors can lead to incorrect simplifications.
  5. Not Simplifying Completely: Make sure you simplify the expression as much as possible. Leaving it partially simplified might not be the answer your teacher or the problem is looking for.

Practice Problems

To really nail this concept, here are a few practice problems you can try:

  1. If f(x) = (x + 3) / (2x - 1), find f(2x).
  2. If f(x) = (4x - 2) / (x + 5), find f(2x).
  3. If f(x) = (5 - x) / (3x + 4), find f(2x).

Work through these problems, and you'll become a pro at finding f(2x) in no time!

Real-World Applications

You might be wondering, "When will I ever use this in real life?" Well, understanding function transformations like this comes in handy in various fields:

  • Engineering: Engineers use function transformations to model and analyze systems. For example, they might use it to understand how changing the input to a system affects its output.
  • Computer Science: In computer graphics and image processing, function transformations are used to scale, rotate, and translate images and objects.
  • Economics: Economists use functions to model economic phenomena, and transformations can help them understand how changes in one variable affect others.
  • Physics: Physicists use function transformations to describe how physical quantities change with respect to time and space.

So, while it might seem abstract now, the concept of function transformations is a fundamental tool in many different areas.

Conclusion

Alright, guys, that wraps up our step-by-step solution to finding f(2x) when given f(x) = (3x - 1) / (x + 2). Remember, the key is to substitute x with 2x carefully and then simplify the expression as much as possible. Avoid those common mistakes, and you'll be golden!

Keep practicing, and you'll become a function transformation master in no time. Happy math-solving!

Additional Tips for Success

Here are some extra tips to help you succeed with these types of problems:

  • Write Neatly: When working through the steps, write neatly and clearly. This will help you avoid making mistakes and make it easier to follow your work.
  • Show Your Work: Always show your work, even if you can do some of the steps in your head. Showing your work makes it easier to catch mistakes and helps your teacher understand your thought process.
  • Check Your Answer: After you've found your answer, take a moment to check it. You can do this by plugging in a value for x into both the original function and your transformed function to see if they match up.
  • Use Online Resources: There are tons of great online resources available to help you with math problems. Websites like Khan Academy and Wolfram Alpha can be invaluable tools.
  • Ask for Help: If you're struggling with a problem, don't be afraid to ask for help. Your teacher, classmates, or online forums can all be great sources of assistance.

Final Thoughts

Mathematics can be challenging, but it's also incredibly rewarding. By breaking down complex problems into smaller, more manageable steps, you can tackle anything that comes your way. So keep practicing, stay curious, and never stop learning!

I hope this guide has been helpful. Good luck with your math endeavors, and remember to have fun while you're at it! You got this!