Inverse Function Of F(x) = (10/9)x + 11: Find It!

Hey guys! Today, we're diving into the fascinating world of inverse functions. Specifically, we're going to tackle the function f(x) = (10/9)x + 11 and figure out what its inverse, denoted as f⁻¹(x), actually is. Don't worry if this sounds intimidating; we'll break it down step by step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding Inverse Functions

Before we jump into the nitty-gritty of our specific function, let's quickly recap what an inverse function really means. Think of a function like a machine: you put something in (the input, x), and it spits something else out (the output, f(x)). An inverse function is like a machine that does the exact opposite. It takes the output of the original function as its input, and it spits out the original input. In mathematical terms, if f(a) = b, then f⁻¹(b) = a. This "undoing" relationship is the core concept behind inverse functions. The key idea is that the inverse function reverses the operation of the original function. For instance, if a function multiplies a number by 2, its inverse will divide the number by 2. This concept of reversal is crucial in various areas of mathematics and its applications, including solving equations and understanding transformations. The inverse function exists only if the original function is bijective, meaning it is both injective (one-to-one) and surjective (onto). In simpler terms, this means that each input has a unique output, and each output has a unique input. If a function is not bijective, we may need to restrict its domain to find an inverse. The notation f⁻¹(x) might look like f raised to the power of -1, but it's important to remember that it represents the inverse function, not the reciprocal. The reciprocal of f(x) would be written as 1/f(x). The process of finding an inverse function involves several algebraic manipulations, such as swapping the roles of x and y and then solving for y. This method ensures that we are effectively reversing the operations performed by the original function. Understanding inverse functions is fundamental for advanced mathematical concepts like logarithms and exponential functions, where inverses play a central role.

Steps to Find the Inverse Function

Okay, now that we've got the basic idea down, let's get practical. Here’s how we find the inverse function for f(x) = (10/9)x + 11. There’s a simple three-step process we can follow, and trust me, it’s not as scary as it sounds! Remember, we're essentially trying to "undo" what the function does.

Step 1: Replace f(x) with y

This is a simple substitution to make the equation easier to work with. So, we rewrite f(x) = (10/9)x + 11 as:

y = (10/9)x + 11

Think of y as just another way of representing the output of the function. This step is more about notational convenience and making the subsequent steps clearer. By replacing f(x) with y, we set the stage for swapping the variables in the next step, which is crucial for finding the inverse. The use of y also aligns with the conventional way of representing functions in a coordinate system, where x is the independent variable and y is the dependent variable. This representation helps in visualizing the function and its inverse graphically, as the graph of the inverse is a reflection of the original function across the line y = x. The simplicity of this substitution belies its importance in the overall process of finding the inverse, as it streamlines the algebraic manipulations that follow. Furthermore, this step reinforces the understanding that a function is a relationship between an input (x) and an output (y), which is fundamental to the concept of inverse functions.

Step 2: Swap x and y

This is the heart of finding the inverse! We're literally switching the roles of the input and output. So, wherever we see a y, we write an x, and wherever we see an x, we write a y. Our equation becomes:

x = (10/9)y + 11

Why do we do this? Because we want to find the function that gives us the original input (x) when we plug in the original output (y). Swapping x and y is the mathematical way of expressing this reversal. This step is the core of the inverse function finding process because it directly implements the concept of reversing the roles of input and output. By interchanging x and y, we are essentially looking at the original function from the opposite perspective. Instead of asking what y is for a given x, we are asking what x was for a given y. This perspective shift is what allows us to find the inverse function. The equation we obtain after swapping x and y represents the inverse relationship, but it is not yet in the standard form of y = f⁻¹(x). The next step, solving for y, is necessary to express the inverse function explicitly. The act of swapping variables highlights the symmetry between a function and its inverse, which is visually represented by their graphs being reflections of each other across the line y = x.

Step 3: Solve for y

Now, we need to isolate y on one side of the equation. This will give us the inverse function in the form y = f⁻¹(x).

Let's walk through the algebra:

1. Subtract 11 from both sides:

   x - 11 = (10/9)y

2. Multiply both sides by 9/10 (the reciprocal of 10/9) to get y by itself:

   (9/10)(x - 11) = y

3. Distribute the 9/10:

   y = (9/10)x - (9/10) * 11

   y = (9/10)x - 99/10

So, the inverse function is:

f⁻¹(x) = (9/10)x - 99/10

Solving for y is the final step in expressing the inverse function in a usable form. This involves using algebraic manipulations to isolate y on one side of the equation, effectively rewriting the equation in the standard y = f⁻¹(x) format. Each algebraic step, such as subtracting 11 from both sides and multiplying by the reciprocal, is designed to undo the operations that were performed on y in the original function. The goal is to peel away the layers of operations until y stands alone, revealing the inverse function. Distributing the 9/10 in the final steps helps to simplify the expression and present it in a clear and understandable format. The result, f⁻¹(x) = (9/10)x - 99/10, is the inverse function we were seeking. This function takes the output of the original function as its input and returns the original input, thus reversing the operation of f(x). The ability to solve for y is a fundamental skill in algebra, and it is crucial for finding inverse functions and manipulating equations in various mathematical contexts.

The Answer

Looking at the options provided, we can see that our inverse function, f⁻¹(x) = (9/10)x - 99/10, matches option C. f⁻¹(x) = (9x - 99)/10. We can rewrite our answer slightly to confirm:

f⁻¹(x) = (9/10)x - 99/10 = (9x/10) - (99/10) = (9x - 99)/10

So, we nailed it! Option C is indeed the correct inverse function. This confirmation step is crucial in any mathematical problem-solving process, as it ensures that the final answer is accurate and aligns with the given options. By comparing the derived inverse function with the provided choices, we can verify that our calculations and algebraic manipulations were correct. In this case, the match with option C reinforces our confidence in the solution. Furthermore, this step highlights the importance of being able to manipulate expressions into different forms, as we needed to rewrite our answer slightly to match one of the options. The ability to simplify and rearrange algebraic expressions is a valuable skill in mathematics, allowing us to express solutions in various equivalent forms. The successful identification of the correct answer demonstrates a solid understanding of the concept of inverse functions and the algebraic techniques required to find them.

Let's Recap

So, guys, finding the inverse of a function might seem tricky at first, but it's all about reversing the operations. We replaced f(x) with y, swapped x and y, and then solved for y. That’s the magic formula! Remember these steps, and you'll be a pro at finding inverse functions in no time. We started by understanding the fundamental concept of inverse functions, which "undo" the operations of the original function. We then applied a three-step process: replacing f(x) with y, swapping x and y, and solving for y. This systematic approach allowed us to find the inverse function f⁻¹(x) = (9/10)x - 99/10. Finally, we verified our answer by matching it with the provided options and confirming that option C was the correct choice. This process not only provides the solution but also reinforces the underlying principles and techniques involved in finding inverse functions. The ability to find inverse functions is a valuable skill in mathematics, with applications in various areas such as calculus, cryptography, and computer science. By mastering this concept, you are equipped to tackle more complex mathematical problems and gain a deeper understanding of mathematical relationships. Keep practicing, and you'll become even more confident in your ability to find inverse functions.

I hope this explanation helped clear things up. Keep practicing, and you'll become a whiz at this in no time! Happy calculating!