How To Find The Inverse Of F(x)=(4x+5)/(2x-2) A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of inverse functions, specifically focusing on how to find the inverse of the function $f(x) = \frac{4x+5}{2x-2}$. Inverse functions are like the "undo" button for functions; they reverse the operation. Understanding how to find them is a fundamental skill in mathematics, especially in algebra and calculus. So, let's break it down step by step and make sure you've got a solid grasp on the process. We'll walk through each stage, explaining the why behind the how, so you're not just memorizing steps, but actually understanding the logic. Get ready to transform functions and impress your friends with your newfound mathematical prowess! Let’s get started!

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

The first step in finding the inverse of a function is to replace the function notation, $f(x)$, with the variable $y$. This might seem like a small change, but it sets us up for the algebraic manipulations we'll be doing next. Think of $y$ as simply another way of representing the output of the function for a given input $x$. So, for our function $f(x) = \frac{4x+5}{2x-2}$, we rewrite it as:

$y = \frac{4x+5}{2x-2}$

This substitution makes the equation easier to work with when we start swapping variables, which is the next crucial step in our journey to find the inverse. It's like switching from a formal name to a nickname – it makes things feel a bit more relaxed and less intimidating! Trust me, this simple step is the key to unlocking the rest of the process. It transforms the functional notation into a standard algebraic equation that we can manipulate. So, embrace the 'y', and let's move on to the next step! Remember, this entire process is about reversing the function's operations, and this initial substitution is our first move in that direction.

Step 2: Swap $x$ and $y$

This is the heart of finding the inverse! We're essentially reversing the roles of the input and output. Wherever you see an $x$, replace it with $y$, and wherever you see a $y$, replace it with $x$. This step reflects the fundamental idea of an inverse function: it takes the output of the original function and turns it back into the original input. For our equation, $y = \frac{4x+5}{2x-2}$, swapping $x$ and $y$ gives us:

$x = \frac{4y+5}{2y-2}$

Now, our goal is to isolate $y$ on one side of the equation. This will give us the inverse function in terms of $x$. It might look a bit intimidating right now with the fractions, but don't worry, we'll tackle it step by step. Think of this swap as a magical transformation, like turning a caterpillar into a butterfly – we're changing the fundamental nature of the equation! This single act of swapping variables is what sets the inverse function apart. It's like looking at the equation in a mirror, reflecting the roles of input and output. So, take a deep breath, and let's move on to the next step – we're getting closer to our goal!

Step 3: Solve for $y$

This is where the algebraic fun begins! Our mission is to isolate $y$ on one side of the equation. This involves a series of algebraic manipulations to get $y$ all by itself. Let's take our equation from the previous step:

$x = \frac{4y+5}{2y-2}$

First, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by the denominator, $(2y-2)$:

$x(2y-2) = 4y+5$

Now, distribute the $x$ on the left side:

$2xy - 2x = 4y + 5$

Next, we want to get all the terms with $y$ on one side and all the other terms on the other side. Let's move the $4y$ to the left and the $2x$ to the right:

$2xy - 4y = 2x + 5$

Now, we can factor out $y$ from the left side:

$y(2x - 4) = 2x + 5$

Finally, to isolate $y$, divide both sides by $(2x - 4)$:

$y = \frac{2x+5}{2x-4}$

We did it! We've successfully solved for $y$. This expression represents the inverse function. This process of isolating $y$ is like untangling a knot – you have to carefully work through each step to reveal the hidden solution. Each algebraic manipulation is a deliberate move, bringing us closer to our goal. Remember, practice makes perfect, so don't be afraid to tackle similar problems to hone your skills!

Step 4: Replace $y$ with $f^{-1}(x)$

The final step is to replace $y$ with the proper notation for the inverse function, which is $f^{-1}(x)$. This notation clearly indicates that we're dealing with the inverse of the original function $f(x)$. So, taking our result from the previous step:

$y = \frac{2x+5}{2x-4}$

We replace $y$ with $f^{-1}(x)$:

$f^{-1}(x) = \frac{2x+5}{2x-4}$

And there you have it! We've found the inverse function. This final step is like putting the finishing touch on a masterpiece – we're formally declaring our result. The notation $f^{-1}(x)$ is a powerful symbol, representing the function that undoes the original $f(x)$. It's a concise way to communicate the inverse relationship. So, celebrate your victory – you've successfully navigated the process of finding an inverse function! This entire journey, from swapping variables to isolating $y$, has equipped you with a valuable skill in the world of functions.

The Inverse Function

So, after all the algebraic maneuvering, we've arrived at our answer. The inverse of the function $f(x) = \frac{4x+5}{2x-2}$ is:

$f^{-1}(x) = \frac{2x+5}{2x-4}$

This means that if you plug a value into $f(x)$ and then plug the result into $f^{-1}(x)$, you should get back your original value (with a few exceptions related to the domain and range, but we won't get into that level of detail here). Think of it like a round trip – you start at one point, go somewhere else, and then come back to where you started. The inverse function is the return ticket! This result is not just a jumble of symbols; it's a powerful tool that allows us to reverse the operation of the original function. It's like having a key that unlocks the original input from the output. So, take a moment to appreciate the elegance of this mathematical concept and the journey we've taken to find this inverse function.

Key Takeaways and Further Practice

Finding the inverse of a function might seem daunting at first, but by following these steps, you can tackle even the trickiest functions. Remember, the key is to:

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve for $y$.
4. Replace $y$ with $f^{-1}(x)$.

These four steps are your roadmap to success! To truly master this skill, practice is essential. Try finding the inverses of other functions, both simple and complex. Look for patterns and shortcuts, and don't be afraid to make mistakes – that's how we learn! Consider functions like linear functions, quadratic functions, and other rational functions. The more you practice, the more comfortable you'll become with the process. You might even start to see the beauty and elegance of inverse functions in the grand scheme of mathematics. So, go forth and conquer the world of inverse functions – you've got this! And remember, understanding inverse functions opens doors to more advanced mathematical concepts, making your journey even more rewarding. Keep exploring, keep learning, and most importantly, keep having fun with math!

I hope this guide has been helpful in your quest to understand inverse functions. If you have any questions, feel free to ask! Happy inverting!