Inverse Function Of F(x) = X/(x-2): A Step-by-Step Guide

by Felix Dubois 57 views

Hey guys! Let's dive into the fascinating world of inverse functions. Today, we're tackling a classic problem: finding the inverse of the function f(x) = x/(x-2). This is a super important concept in mathematics, and understanding it can unlock a whole new level of problem-solving skills. So, grab your thinking caps, and let's get started!

What are Inverse Functions?

Before we jump into the nitty-gritty of this specific problem, let's quickly recap what inverse functions are all about. Think of a function like a machine: you feed it an input (x), and it spits out an output (y or f(x)). An inverse function is like that machine in reverse! It takes the output (y) and spits back the original input (x).

Formally speaking, if we have a function f(x), its inverse, denoted as f⁻¹(x), satisfies these two key properties:

  • f⁻¹(f(x)) = x for all x in the domain of f.
  • f(f⁻¹(x)) = x for all x in the domain of f⁻¹.

In simpler terms, if you plug x into f and then plug the result into f⁻¹, you get x back. The same goes if you do it the other way around. This "undoing" relationship is the heart and soul of inverse functions.

Why are inverse functions important, you ask? Well, they pop up everywhere in math and its applications! They're crucial for solving equations, understanding relationships between variables, and even in fields like cryptography and computer science. Mastering inverse functions is like adding a powerful tool to your mathematical toolkit.

The Step-by-Step Guide to Finding Inverse Functions

Alright, now that we've got the basics down, let's get practical. Finding the inverse of a function involves a few straightforward steps. Let's break them down:

  1. Replace f(x) with y: This is simply a notational change to make the algebra a bit easier to handle. Our function f(x) = x/(x-2) becomes y = x/(x-2).
  2. Swap x and y: This is the key step in finding the inverse! We're essentially reversing the roles of input and output. Our equation now becomes x = y/(y-2).
  3. Solve for y: This is where the algebraic magic happens. We need to isolate y on one side of the equation. This might involve some clever manipulation, but don't worry, we'll walk through it together.
  4. Replace y with f⁻¹(x): Once we've solved for y, we replace it with the inverse function notation f⁻¹(x). This tells us that we've found the inverse function.

These four steps are your roadmap to finding any inverse function. Keep them in mind as we tackle our specific problem. Remember, practice makes perfect, so the more you work through these steps, the more comfortable you'll become.

Cracking the Code: Finding the Inverse of f(x) = x/(x-2)

Okay, let's put our newfound knowledge to the test and find the inverse of f(x) = x/(x-2). We'll follow our step-by-step guide and see how it all comes together.

Step 1: Replace f(x) with y

As we discussed, the first step is a simple change in notation: y = x/(x-2). Nothing too tricky here!

Step 2: Swap x and y

Now comes the crucial swap: x = y/(y-2). We've effectively reversed the roles of input and output. This is the heart of finding the inverse.

Step 3: Solve for y

This is where things get a little more interesting. Our goal is to isolate y. Here's how we can do it:

  1. Multiply both sides by (y-2): This gets rid of the fraction: x(y-2) = y
  2. Distribute the x: xy - 2x = y
  3. Get all y terms on one side: xy - y = 2x
  4. Factor out y: y(x - 1) = 2x
  5. Divide both sides by (x-1): y = 2x/(x-1)

Step 4: Replace y with f⁻¹(x)

We've done the hard work! Now we replace y with the inverse function notation: f⁻¹(x) = 2x/(x-1).

And there you have it! The inverse function of f(x) = x/(x-2) is f⁻¹(x) = 2x/(x-1). Give yourself a pat on the back!

The Answer and Why It's Important

So, looking back at our options, the correct answer is D. f⁻¹(x) = 2x/(x-1). Awesome! We nailed it.

But let's not stop there. It's crucial to understand why this is the correct answer and what it means. Remember our definition of inverse functions? f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. We can actually verify our answer by plugging it back into the original function and seeing if we get x.

Let's try it out. We'll plug f⁻¹(x) = 2x/(x-1) into f(x) = x/(x-2):

f(f⁻¹(x)) = f(2x/(x-1)) = (2x/(x-1)) / ((2x/(x-1)) - 2)

This looks a bit messy, but let's simplify it:

  • Multiply the numerator and denominator by (x-1): f(f⁻¹(x)) = 2x / (2x - 2(x-1)).
  • Simplify the denominator: f(f⁻¹(x)) = 2x / (2x - 2x + 2).
  • Further simplification gives us: f(f⁻¹(x)) = 2x / 2 = x.

Hooray! It works! This confirms that we've indeed found the correct inverse function.

Common Pitfalls and How to Avoid Them

Finding inverse functions can be tricky, and there are a few common mistakes that students often make. Let's highlight these pitfalls and how to avoid them:

  1. Forgetting to swap x and y: This is the most crucial step! If you don't swap x and y, you're not finding the inverse. Double-check that you've done this step correctly.
  2. Algebraic errors when solving for y: Solving for y can involve several algebraic manipulations, and it's easy to make a mistake. Take your time, write out each step clearly, and double-check your work.
  3. Not considering the domain and range: The domain and range of a function and its inverse are closely related. The domain of f is the range of f⁻¹, and vice versa. Be mindful of any restrictions on the domain or range, especially when dealing with fractions or square roots.
  4. Confusing f⁻¹(x) with 1/f(x): This is a common mistake! f⁻¹(x) represents the inverse function, while 1/f(x) represents the reciprocal of the function. They are not the same thing!

By being aware of these common pitfalls, you can avoid making these mistakes and confidently find inverse functions.

Wrapping Up: Inverse Functions Mastered!

Alright, guys, we've covered a lot of ground! We've explored the concept of inverse functions, learned the step-by-step process for finding them, and tackled a specific example: finding the inverse of f(x) = x/(x-2). We even discussed common pitfalls and how to avoid them. You're now well-equipped to tackle inverse function problems with confidence!

Remember, practice is key to mastering any mathematical concept. So, try working through more examples, and don't hesitate to ask for help if you get stuck. The more you practice, the more natural these steps will become. Keep up the great work, and happy inverting!