Solving For 'a': F(a) = F(3) - A Explained!

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Hey guys! Let's dive into solving a cool math problem together. We've got a function, f(x) = 2x + 1, and our mission is to find the real number 'a' that satisfies the equation f(a) = f(3) - a. Don’t worry; we’ll break it down step by step, making it super easy to follow. So, grab your favorite beverage, and let’s get started!

Understanding the Problem

At its core, this problem involves function evaluation and solving a simple algebraic equation. We're given a linear function, f(x) = 2x + 1, which means for any input 'x', the function doubles it and adds 1. The challenge is to find a specific value 'a' such that when we plug it into the function, it equals the result of f(3) minus 'a'.

To kick things off, it’s important to understand what each part of the equation means. f(a) means we're plugging 'a' into our function, and f(3) means we're plugging in 3. The equation f(a) = f(3) - a sets up a relationship we need to solve. We’re essentially looking for an 'a' that makes this equation true. This involves a few key steps: first, we need to calculate f(3). Then, we'll substitute 'a' into the function to get f(a). Finally, we'll set up and solve the equation.

This type of problem often pops up in algebra and pre-calculus courses, so mastering it will definitely help you out. Remember, the beauty of math lies in its systematic approach. By breaking down complex problems into smaller, manageable steps, we can tackle anything! So, let’s roll up our sleeves and get into the solution. We’ll take it one step at a time, ensuring everyone understands each part. Ready? Let's go!

Step 1: Calculate f(3)

Alright, the first step in our mathematical journey is to calculate f(3). Remember, our function is f(x) = 2x + 1. So, to find f(3), we simply replace 'x' with 3 in the function. This is a fundamental concept in function evaluation, and it's super important to get it right.

So, we have:

f(3) = 2 * (3) + 1

Now, let's do the math. First, we multiply 2 by 3, which gives us 6. Then, we add 1 to the result:

f(3) = 6 + 1 f(3) = 7

Therefore, f(3) equals 7. This is a crucial piece of information that we'll use later in the equation. Calculating f(3) is like setting the stage for the main act. We’ve now got a numerical value for one part of our equation, which simplifies things quite a bit.

This step illustrates the basic process of function evaluation. Whenever you see f with a number inside the parentheses, it’s telling you to plug that number into the function wherever you see 'x'. It’s like a machine: you put in an input (in this case, 3), and the function spits out an output (in this case, 7). Understanding this concept is essential for tackling more complex function-related problems. So, now that we've successfully calculated f(3), we're one step closer to solving our original problem. Let’s move on to the next step, where we’ll deal with f(a).

Step 2: Determine f(a)

Now that we've figured out f(3), let's tackle f(a). Remember, our function is f(x) = 2x + 1. Finding f(a) is just like finding f(3), but instead of plugging in 3, we're plugging in 'a'. This means we replace 'x' with 'a' in the function.

So, we have:

f(a) = 2 * (a) + 1

This simplifies to:

f(a) = 2a + 1

Thus, f(a) is equal to 2a + 1. This expression represents the value of the function when 'a' is the input. We're not getting a single number here like we did with f(3) because 'a' is a variable. Instead, we have an algebraic expression.

This step is important because it highlights how functions work with variables. Functions aren't just for numbers; they can take variables as inputs too. When we plug in a variable, the output is often an algebraic expression that involves that variable. In this case, plugging in 'a' resulted in the expression 2a + 1. This expression will be crucial in the next step when we set up our equation. So, we've now determined both f(3) and f(a). We're building up all the pieces we need to solve the puzzle. Next up, we'll put these pieces together and actually solve for 'a'. Let’s get to it!

Step 3: Set Up the Equation

Okay, we’ve done the groundwork – we know f(3) = 7 and f(a) = 2a + 1. Now it’s time to set up the main equation that we were given in the problem: f(a) = f(3) - a. This equation is the key to finding the value of 'a'.

We'll substitute the values we calculated into the equation. We know f(a) = 2a + 1 and f(3) = 7, so we can rewrite the equation as:

2a + 1 = 7 - a

This is our equation! It’s a simple linear equation with 'a' as the variable. Setting up the equation correctly is super important because if we mess this up, the rest of the solution will be incorrect. Make sure you double-check that you've substituted the correct values in the right places.

This step bridges the gap between function evaluation and equation solving. We've taken the functional notation and translated it into a standard algebraic equation. This is a common technique in math – we often use what we know about functions to create equations that we can solve. Now that we have our equation, the next step is to solve it for 'a'. This involves using basic algebraic manipulations to isolate 'a' on one side of the equation. Are you ready to solve for 'a'? Let’s move on!

Step 4: Solve for 'a'

Alright, we've got our equation: 2a + 1 = 7 - a. Now, let's solve for 'a'. This involves using basic algebraic principles to isolate 'a' on one side of the equation. Our goal is to get 'a' by itself so we can see its value.

The first thing we want to do is get all the 'a' terms on one side of the equation. To do this, we can add 'a' to both sides. This keeps the equation balanced:

2a + 1 + a = 7 - a + a

This simplifies to:

3a + 1 = 7

Great! Now we have all the 'a' terms on the left side. Next, we want to get rid of the +1 on the left side. We can do this by subtracting 1 from both sides:

3a + 1 - 1 = 7 - 1

This simplifies to:

3a = 6

We're almost there! Now we have 3a = 6. To isolate 'a', we need to divide both sides of the equation by 3:

3a / 3 = 6 / 3

This gives us:

a = 2

So, we've found our answer! a = 2. This means that the real number 'a' that satisfies the equation f(a) = f(3) - a is 2.

Solving for 'a' is a classic example of how algebraic manipulation works. We used addition, subtraction, and division to move terms around and isolate the variable we were interested in. Remember, the key is to do the same operation on both sides of the equation to keep it balanced. Now that we've found 'a', let's do one more important step to make sure our answer is correct. Let’s verify our solution!

Step 5: Verify the Solution

We've found that a = 2. But before we celebrate, let's verify our solution to make sure it's correct. This is a crucial step in problem-solving because it helps us catch any mistakes we might have made along the way. To verify, we'll plug a = 2 back into our original equation, f(a) = f(3) - a, and see if it holds true.

First, let's calculate f(2) using our function f(x) = 2x + 1:

f(2) = 2 * (2) + 1 f(2) = 4 + 1 f(2) = 5

Now, let's plug in a = 2 and f(3) = 7 into the right side of our original equation:

f(3) - a = 7 - 2 f(3) - a = 5

So, we have f(2) = 5 and f(3) - a = 5. This means that both sides of the equation are equal when a = 2:

5 = 5

Our equation holds true! This confirms that our solution, a = 2, is indeed correct. Verification is like the final seal of approval on our solution. It gives us confidence that we've not only found an answer but that it's the right answer.

This step demonstrates the importance of checking your work. In math, it’s not just about getting an answer; it’s about making sure that answer is correct. By plugging our solution back into the original equation, we’ve provided solid proof that we’ve solved the problem correctly. Now that we've verified our solution, we can confidently say that we've mastered this problem! Let’s wrap things up with a summary.

Conclusion

Awesome job, guys! We successfully found the real number 'a' for which f(a) = f(3) - a, given the function f(x) = 2x + 1. We broke down the problem into manageable steps:

  1. Calculated f(3): We found that f(3) = 7.
  2. Determined f(a): We expressed f(a) as 2a + 1.
  3. Set up the equation: We substituted our values into the equation f(a) = f(3) - a to get 2a + 1 = 7 - a.
  4. Solved for 'a': We used algebraic manipulation to find that a = 2.
  5. Verified the solution: We plugged a = 2 back into the original equation and confirmed that it holds true.

This problem was a fantastic exercise in function evaluation and solving linear equations. Remember, the key to tackling math problems is to break them down into smaller, more manageable steps. By systematically working through each step, we can solve even the trickiest problems.

Keep practicing these types of problems, and you'll become a math whiz in no time! If you ever get stuck, remember to revisit these steps and take it one piece at a time. You've got this! Keep up the great work, and happy math-solving!