Finding The Inverse Function Value $f^{-1}(-2)$ Given $f(x) = rac{2-36}{2x-4}$

Hey guys! 👋 Ever stumbled upon a math problem that looks like a tangled mess of fractions and variables? Don't worry, we've all been there! Today, we're going to break down a classic inverse function problem. We're given a function $f(x)$ and asked to find the value of its inverse at a specific point. Sounds intimidating? Trust me, it's not as scary as it seems. We'll take it step by step, making sure you understand the logic behind each move. So, buckle up and let's dive into the fascinating world of inverse functions! 🚀

Understanding the Problem

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

$f(x) = \frac{2-36}{2x-4}$

Okay, first things first, let's simplify that fraction. It looks a little cluttered with 2-36 in the numerator. Let's take care of that:

$f(x) = \frac{-34}{2x-4}$

Much cleaner, right? Now, we need to find $f^{-1}(-2)$. Remember, $f^{-1}(x)$ represents the inverse function of $f(x)$. Basically, if $f(a) = b$, then $f^{-1}(b) = a$. So, we're looking for the value of $x$ that, when plugged into the inverse function, gives us -2. But wait, we don't have the inverse function explicitly! No problem, we can find it. Finding the inverse function is like reversing a recipe – we need to undo all the operations performed by the original function. We aim to find the input value that corresponds to an output of -2 for the inverse function. This is a classic problem in algebra and calculus, and it's a great way to test your understanding of function inverses. It's also a foundational concept that comes up in more advanced math topics, so mastering this now will definitely pay off later! Think of it like building a strong base for a mathematical skyscraper – the higher you want to go, the sturdier the base needs to be.

Key Concepts to Remember:

• Inverse Function: A function that "undoes" the original function. If $f(a) = b$, then $f^{-1}(b) = a$.
• Notation: $f^{-1}(x)$ represents the inverse of the function $f(x)$.
• Goal: Find the value of $x$ such that $f^{-1}(-2)$ is satisfied.

Finding the Inverse Function

The key to finding the inverse function is to swap $x$ and $y$ (where $y = f(x)$) and then solve for $y$. Let's do it! 💪

1. Replace $f(x)$ with $y$:

   $y = \frac{-34}{2x-4}$

   This step is simply a change in notation to make the following steps clearer. We're just replacing the function notation with a single variable to make the algebraic manipulation easier. It's like switching from writing out a full sentence to using shorthand – same meaning, but quicker to write! Think of $y$ as the output of the function, and $x$ as the input. Our goal now is to rewrite the equation so that $y$ is the input and $x$ is the output – that's the essence of finding the inverse.

2. Swap $x$ and $y$:

   $x = \frac{-34}{2y-4}$

   This is the crucial step where we actually perform the inversion. We're reflecting the function across the line $y = x$. In other words, we're interchanging the roles of input and output. This is the heart of the inverse function concept. It's like looking at a reflection in a mirror – the left and right sides are swapped. Similarly, the input and output of the function are swapped when we find the inverse.

3. Solve for $y$:

   This is where the algebra comes in! We need to isolate $y$ on one side of the equation. This might involve multiple steps, like multiplying both sides, adding or subtracting terms, and dividing. Think of it like untangling a knot – we need to carefully undo each step that was used to tie the knot in the first place.

   • Multiply both sides by $(2y-4)$:

     $x(2y-4) = -34$

     We do this to get rid of the fraction. It's like clearing the denominator – we want to work with a simpler equation.

   • Distribute $x$:

     $2xy - 4x = -34$

     This step expands the expression on the left side, making it easier to isolate the term with $y$.

   • Add $4x$ to both sides:

     $2xy = 4x - 34$

     We're moving the term without $y$ to the right side of the equation.

   • Divide both sides by $2x$:

     $y = \frac{4x - 34}{2x}$

     Finally, we have isolated $y$! This is our inverse function.

4. Simplify (optional):

   We can simplify the expression by dividing both the numerator and denominator by 2:

   $y = \frac{2x - 17}{x}$

   Simplifying makes the function easier to work with. It's like tidying up your workspace – a cleaner function is less prone to errors.

5. Replace $y$ with $f^{-1}(x)$:

   $f^{-1}(x) = \frac{2x - 17}{x}$

   This is the final step in finding the inverse function. We're replacing $y$ with the proper notation for the inverse function.

   Therefore, the inverse function is:

   $f^{-1}(x) = \frac{2x - 17}{x}$

   Woohoo! We found the inverse function! 🎉 This was a big step, so take a moment to pat yourself on the back. You've successfully navigated the tricky process of finding an inverse function. Now, let's use this to solve the original problem.

Evaluating the Inverse Function

Now that we have the inverse function, we can find $f^{-1}(-2)$. This is the easy part! We simply plug in -2 for $x$ in the inverse function we just found:

$f^{-1}(-2) = \frac{2(-2) - 17}{-2}$

Let's simplify this expression:

$f^{-1}(-2) = \frac{-4 - 17}{-2}$

$f^{-1}(-2) = \frac{-21}{-2}$

$f^{-1}(-2) = \frac{21}{2}$

Therefore, $f^{-1}(-2) = \frac{21}{2}$

And that's it! We found the value of the inverse function at -2. Give yourself a round of applause! 👏 You've successfully tackled a problem involving inverse functions from start to finish. Remember, the key is to break down the problem into smaller, manageable steps. Don't be afraid to take your time and work through each step carefully. And most importantly, practice makes perfect! The more you work with inverse functions, the more comfortable you'll become with them.

Conclusion

So, guys, we've conquered another math challenge! We started with a function, found its inverse, and then evaluated the inverse at a specific point. We've seen how swapping variables and solving for $y$ allows us to "undo" the original function. Remember, practice is key to mastering these concepts. Try working through similar problems on your own, and don't hesitate to ask for help if you get stuck. Math can be challenging, but it's also incredibly rewarding when you finally crack a tough problem. Keep exploring, keep learning, and keep those math muscles flexing! 💪 Remember, every problem you solve is a step closer to becoming a math whiz! Keep up the awesome work! ✨