Finding F(f⁻¹(3)): A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem: finding the value of the composite function $f(f^{-1}(3))$. Don't let the notation scare you; it's much simpler than it looks! We'll be using a table of values for the function f(x) to figure this out. So, buckle up, and let's get started!

Understanding the Problem: What are we looking for?

Before we jump into calculations, let's break down what $f(f^{-1}(3))$ actually means. This is a composition of functions, which essentially means we're plugging one function into another. In this case, we're plugging the inverse of the function f, denoted as $f^{-1}$, into the function f itself. The input to the inverse function is 3. So, the expression can be read as "f of the inverse of f of 3".

The key here is understanding what an inverse function does. If f(x) = y, then $f^{-1}(y) = x$. In simpler terms, the inverse function "undoes" what the original function does. If f takes x to y, then $f^{-1}$ takes y back to x. This is a crucial concept for solving our problem.

Now, consider $f(f^{-1}(3))$. Let's say $f^{-1}(3) = a$. Then, what we're really trying to find is f(a). But remember, since $f^{-1}(3) = a$, that means f(a) = 3. This is because the inverse function essentially reverses the mapping of the original function. Therefore, the composition of a function and its inverse simply results in the original input. In mathematical notation, $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all x in the domain where the functions are defined. This holds true as long as 3 is in the range of f and the inverse function is defined at 3.

Given this knowledge, you might already see the answer! But let's walk through the process step-by-step using the provided table to solidify our understanding. This will be particularly helpful when we encounter more complex scenarios or situations where we don't have this neat inverse property to rely on directly. Understanding the underlying principles is more important than just memorizing a trick.

Using the Table: Finding the Value

We have the following table:

Our goal is to find $f(f^{-1}(3))$. The first step is to determine the value of $f^{-1}(3)$. Remember, $f^{-1}(3)$ is the value of x that makes f(x) = 3. So, we need to look at the table and find where the f(x) column has the value 3.

Looking at the table, we see that when x = 0, f(x) = 3. This means $f(0) = 3$. Therefore, by the definition of the inverse function, $f^{-1}(3) = 0$. Isn't that neat? We found the value of the inverse function using the table.

Now that we know $f^{-1}(3) = 0$, we can substitute this back into our original expression: $f(f^{-1}(3)) = f(0)$. So, our next step is to find the value of f(0). Again, we can use the table for this. We simply look for the row where x = 0 and read the corresponding value of f(x).

From the table, we see that when x = 0, f(x) = 3. Therefore, f(0) = 3. So, after all this, we've determined that $f(f^{-1}(3)) = 3$.

The Quick Way: Using the Inverse Property

Now, let's remember the property we discussed earlier: $f(f^{-1}(x)) = x$. This property tells us that if we apply a function and then its inverse (or vice versa), we end up back where we started. In our case, we have $f(f^{-1}(3))$. According to the property, this should simply equal 3.

Notice how this matches the answer we got by working through the problem step-by-step using the table. This property can be a huge time-saver when you're dealing with composite functions and their inverses. However, it's always good to understand the underlying principles and be able to solve the problem using different methods, especially in situations where the property might not be directly applicable.

Key Takeaways: What did we learn?

Let's recap the key concepts we covered in this problem:

• Inverse Functions: An inverse function "undoes" the action of the original function. If f(x) = y, then $f^{-1}(y) = x$.
• Composition of Functions: This involves plugging the output of one function into another.
• The Inverse Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ (when the functions are defined).
• Using Tables: We can use tables of values to find the outputs of functions and their inverses.

Understanding these concepts will help you tackle a wide range of math problems involving functions and their inverses.

Let's Practice!: Try these problems.

To solidify your understanding, try solving similar problems. Here are a couple of examples:

1. Using the table above, find $f^{-1}(0)$.
2. Using the table above, find $f(f^{-1}(-4))$.
3. If $g(x) = 2x + 1$, find $g^{-1}(x)$ and then find $g(g^{-1}(5))$.

Working through these practice problems will help you master the concepts we've discussed and build your problem-solving skills. Don't be afraid to make mistakes; they're a valuable part of the learning process!

Conclusion: You've got this!

So, there you have it! We've successfully found the value of $f(f^{-1}(3))$ using both a step-by-step approach with the table and the handy inverse property. Remember the key concepts, practice regularly, and you'll be a pro at solving these types of problems in no time. Keep up the great work, guys, and happy problem-solving!