Inverse Function: F(x) = (9 + 2x) / (10 - X)

Alright, guys, let's dive into finding the inverse of a function. Specifically, we're tackling the function f(x) = (9 + 2x) / (10 - x). This is a classic problem in mathematics, and understanding how to find inverse functions is super useful. So, let's break it down step by step.

Understanding Inverse Functions

Before we jump into the nitty-gritty, let's quickly recap what an inverse function actually is. Essentially, if f(x) takes an input x and gives you an output y, the inverse function, denoted as f^{-1}(x), does the opposite. It takes y as an input and spits out x. Think of it as undoing what the original function did. Mathematically, this means f^{-1}(f(x)) = x and f(f^{-1}(x)) = x. Knowing this fundamental concept is crucial because it guides our entire process. When we look at f(x), we are essentially mapping from the domain to the range, while f^{-1}(x) maps us back from the range to the domain. Therefore, finding the inverse involves a bit of algebraic manipulation to switch the roles of x and y and then solve for the new y. This process requires a solid understanding of algebraic principles, including how to manipulate equations, isolate variables, and simplify expressions. The goal here is not just to find a solution but also to understand the underlying relationship between a function and its inverse. Remember, not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning each x-value corresponds to a unique y-value and vice versa. This is also known as the horizontal line test. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse. In the case of rational functions like the one we're dealing with, we often need to consider the domain and range to ensure the inverse is properly defined. So, let's keep these concepts in mind as we proceed with the steps to find the inverse of f(x) = (9 + 2x) / (10 - x).

Step 1: Replace f(x) with y

The first thing we're gonna do is replace f(x) with y. This just makes the equation a little easier to work with. So, we rewrite our function as:

y = (9 + 2x) / (10 - x)

This simple substitution is just a notational change but helps to clarify the relationship between x and y. By replacing f(x) with y, we're setting the stage to swap the roles of x and y in the next step, which is the heart of finding the inverse. It’s like saying, “Okay, instead of thinking of this as a function that takes x and gives us y, let’s think about what happens if we treat y as the input and try to find x.” This small change makes it clearer that we're about to reverse the process. Furthermore, using y simplifies the algebraic manipulations in the following steps. It reduces visual clutter and makes it easier to keep track of the terms as we rearrange and solve the equation. Think of it as preparing the canvas before painting – it’s a small step, but it sets the foundation for the rest of the process. So, with this substitution in place, we're ready to move on to the next crucial step: swapping x and y to start unraveling the original function.

Step 2: Swap x and y

Now comes the fun part where we switch x and y. This is the core step in finding the inverse because we're literally reversing the roles of input and output. Our equation now looks like this:

x = (9 + 2y) / (10 - y)

What we've done here is essentially turned the original function inside out. Instead of y being expressed in terms of x, we now have x expressed in terms of y. This swap is the key to unlocking the inverse function. It reflects the fundamental idea that the inverse function undoes what the original function does. It's important to remember that we're not just changing letters; we're changing the entire perspective. The original function takes an x-value, does some math to it, and gives us a y-value. The inverse function takes that y-value and, through a different set of operations, gets us back to the original x-value. By swapping x and y, we're setting up the equation so that we can solve for y in terms of x. This will give us the formula for the inverse function, f^{-1}(x). So, this step is not just a simple algebraic trick; it's a conceptual shift that allows us to see the function from a reversed point of view. With x and y swapped, we're now ready to isolate y and express it as a function of x, which will be our inverse function.

Step 3: Solve for y

Okay, time to roll up our sleeves and isolate y. This is where the algebra skills come into play. We need to get y all by itself on one side of the equation.

Starting with:

x = (9 + 2y) / (10 - y)

First, multiply both sides by (10 - y) to get rid of the fraction:

x(10 - y) = 9 + 2y

Expand the left side:

10x - xy = 9 + 2y

Now, let's get all the terms with y on one side and everything else on the other side:

10x - 9 = 2y + xy

Factor out y from the right side:

10x - 9 = y(2 + x)

Finally, divide by (2 + x) to solve for y:

y = (10x - 9) / (2 + x)

This is the most algebraically intensive part of the process, and it requires careful attention to detail to avoid errors. Each step involves manipulating the equation while maintaining its balance. Multiplying by (10 - y) clears the fraction, making the equation easier to work with. Expanding the left side allows us to separate the terms containing y. Rearranging the terms to group all y-terms on one side is crucial for factoring out y. Factoring out y allows us to isolate it by dividing by the remaining expression. This entire process relies on the fundamental principles of algebra, such as the distributive property, combining like terms, and performing inverse operations. It’s a step-by-step transformation of the equation until we have y expressed explicitly in terms of x. The result, y = (10x - 9) / (2 + x), represents the inverse function we've been searching for. Now, we just need to replace y with the proper notation for the inverse function to complete the process.

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

We're almost there! The last step is to replace y with f^{-1}(x) to show that we've found the inverse function:

f^{-1}(x) = (10x - 9) / (2 + x)

And that's it! We found the inverse function. Replacing y with f^{-1}(x) is the final touch that signifies we have successfully found the inverse. This notation clearly indicates that the function we've derived is the inverse of the original function f(x). It's not just any function; it's the one that undoes what f(x) does. This notation is important because it distinguishes the inverse function from other functions and makes it clear that it has a specific relationship with f(x). In essence, we're putting the final stamp on our work. This notation also helps to communicate the result clearly to others. When someone sees f^{-1}(x) = (10x - 9) / (2 + x), they immediately know that this is the inverse of the original function f(x) = (9 + 2x) / (10 - x). It's a concise and universally understood way to express the relationship between a function and its inverse. So, by making this final substitution, we're not just completing the algebra; we're also ensuring that our result is clearly and accurately communicated.

Conclusion

So, to wrap things up, the inverse of the function f(x) = (9 + 2x) / (10 - x) is f^{-1}(x) = (10x - 9) / (2 + x). Remember, finding the inverse involves swapping x and y and then solving for y. This is a fundamental concept in math, and I hope this explanation made it a bit clearer for you guys!

Finding the inverse of a function is a valuable skill in mathematics, with applications in various fields such as calculus, cryptography, and computer science. Understanding the process not only helps in solving mathematical problems but also enhances analytical and problem-solving abilities. Keep practicing, and you'll become a pro at finding inverse functions in no time!