Solving F⁻¹(f⁻¹(f⁻¹(1))) For F(x) = 2x + 3: A Detailed Solution

Hey everyone! Today, we're diving into a cool math problem that involves inverse functions. It might look a little intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. Our mission? To figure out the value of f⁻¹(f⁻¹(f⁻¹(1))) when f(x) is defined as 2x + 3. We have some options to choose from: a) 0, b) -1, c) -2, or d) -3. Let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We're given a function, f(x) = 2x + 3, and we need to find the value of f⁻¹(f⁻¹(f⁻¹(1))). This notation might seem a bit confusing, but it's all about inverse functions. So, what exactly is an inverse function?

An inverse function, denoted as f⁻¹(x), essentially reverses what the original function does. Think of it like this: if f(x) takes an input x and transforms it into an output y, then f⁻¹(x) takes that output y and transforms it back into the original input x. In simpler terms, it undoes the operation of the original function. For example, if f(x) adds 3 to a number, then f⁻¹(x) will subtract 3 from that number.

In our problem, we need to apply the inverse function three times, starting with the input 1. This means we first need to find f⁻¹(1), then apply the inverse function to that result, and finally, apply the inverse function one more time. It's like peeling an onion, one layer at a time. Each application of the inverse function gets us closer to the final answer. So, let's roll up our sleeves and figure out how to find the inverse function and solve this problem.

Finding the Inverse Function

The first crucial step in solving our problem is to find the inverse function, f⁻¹(x), of the given function f(x) = 2x + 3. There’s a straightforward method to do this, which involves a few simple algebraic manipulations. Let's walk through it together.

1. Replace f(x) with y: This makes the equation easier to work with. So, we rewrite f(x) = 2x + 3 as y = 2x + 3.
2. Swap x and y: This is the key step in finding the inverse. We interchange x and y, resulting in the equation x = 2y + 3. This step reflects the idea that the inverse function reverses the roles of input and output.
3. Solve for y: Now, we need to isolate y on one side of the equation. First, subtract 3 from both sides: x - 3 = 2y. Then, divide both sides by 2: (x - 3) / 2 = y. This gives us y = (x - 3) / 2.
4. Replace y with f⁻¹(x): Finally, we replace y with the inverse function notation f⁻¹(x). This gives us our inverse function: f⁻¹(x) = (x - 3) / 2.

Now that we've found the inverse function, f⁻¹(x) = (x - 3) / 2, we're ready to tackle the original problem. Remember, we need to find f⁻¹(f⁻¹(f⁻¹(1))). This means we'll be using the inverse function three times, so let's take it one step at a time.

Step-by-Step Solution

Now that we've successfully found the inverse function, f⁻¹(x) = (x - 3) / 2, we can finally solve the main problem: finding the value of f⁻¹(f⁻¹(f⁻¹(1))). Remember, this means we need to apply the inverse function three times, starting with the input 1. Let's break it down step by step to make it crystal clear.

Step 1: Find f⁻¹(1)

First, we need to find the value of the inverse function when the input is 1. This means we substitute x = 1 into our inverse function: f⁻¹(1) = (1 - 3) / 2. Let's simplify this:

• 1 - 3 = -2
• -2 / 2 = -1

So, f⁻¹(1) = -1. Great! We've completed the first step.

Step 2: Find f⁻¹(f⁻¹(1))

Next, we need to find f⁻¹(f⁻¹(1)). We already know that f⁻¹(1) = -1, so we can rewrite this as f⁻¹(-1). Now, we substitute x = -1 into our inverse function:

• f⁻¹(-1) = (-1 - 3) / 2

Let's simplify this:

• -1 - 3 = -4
• -4 / 2 = -2

So, f⁻¹(f⁻¹(1)) = f⁻¹(-1) = -2. We're making progress!

Step 3: Find f⁻¹(f⁻¹(f⁻¹(1)))

Finally, we need to find f⁻¹(f⁻¹(f⁻¹(1))). We know that f⁻¹(f⁻¹(1)) = -2, so we can rewrite this as f⁻¹(-2). Now, we substitute x = -2 into our inverse function:

• f⁻¹(-2) = (-2 - 3) / 2

Let's simplify this:

• -2 - 3 = -5
• -5 / 2 = -2.5

So, f⁻¹(f⁻¹(f⁻¹(1))) = f⁻¹(-2) = -2.5. But wait a minute! This answer, -2.5, isn't one of the options given in the problem. It seems there might be a slight mistake in the calculation or the options provided. Let’s double-check our steps to ensure accuracy.

Double-Checking Our Work

Okay, let's take a deep breath and carefully go through our calculations again to make sure we haven't made any sneaky mistakes. It’s always a good idea to double-check your work, especially in math problems, to avoid silly errors.

Recalculating Step 3

We found that f⁻¹(-2) = (-2 - 3) / 2. Let's re-evaluate this:

• -2 - 3 = -5. This part is correct.
• -5 / 2 = -2.5. This is also correct.

So, our calculation of f⁻¹(-2) as -2.5 is indeed accurate. This means that the correct answer, -2.5, is not among the options provided (a) 0, b) -1, c) -2, d) -3. It's possible that there was a mistake in the question itself or the provided answer choices. This can happen sometimes, and it's important to be able to recognize when your solution doesn't match the given options.

In a real-world scenario, if you encountered this on a test or assignment, you might want to politely ask your teacher or professor to double-check the question and the answer choices. It's always better to clarify than to force your answer into an incorrect option.

Conclusion

In summary, we set out to find the value of f⁻¹(f⁻¹(f⁻¹(1))) for the function f(x) = 2x + 3. We successfully found the inverse function, f⁻¹(x) = (x - 3) / 2, and then applied it three times, step by step. We calculated:

• f⁻¹(1) = -1
• f⁻¹(f⁻¹(1)) = -2
• f⁻¹(f⁻¹(f⁻¹(1))) = -2.5

However, our final answer, -2.5, doesn't match any of the given options. This suggests a potential error in the question or the provided answer choices. While it's a bit unsatisfying not to have a matching answer, we've demonstrated a solid understanding of inverse functions and how to apply them. Great job, guys! Remember, the process is just as important as the final answer. Keep practicing, and you'll become a math whiz in no time!