Finding The Inverse Function Of F(x) = (1/9)x + 2
In mathematics, particularly in algebra, understanding the concept of inverse functions is crucial. The inverse of a function essentially 'undoes' what the original function does. In this article, we will delve into the process of finding the inverse of a linear function, using the example of f(x) = (1/9)x + 2. We will break down each step, making it easy to understand and apply to other similar problems. Whether you are a student learning about functions or someone looking to refresh your math skills, this guide will provide a clear and concise explanation. So, let's embark on this mathematical journey and unravel the mystery of inverse functions.
Understanding Inverse Functions
Before we jump into the specific problem, let's first grasp the fundamental concept of inverse functions. Think of a function as a machine that takes an input, processes it, and produces an output. The inverse function is another machine that takes the output of the original function and returns the original input. In mathematical terms, if we have a function f(x), its inverse is denoted as f⁻¹(x). The key property of inverse functions is that if you apply a function and then its inverse (or vice versa), you end up with the original input. This can be expressed as f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. Understanding this concept is crucial because it lays the foundation for the steps we will take to find the inverse of a given function. It's not just about mechanically following steps; it's about understanding the underlying principle of reversing the function's operation. For example, if a function multiplies a number by 2 and then adds 3, the inverse function will subtract 3 and then divide by 2. This intuitive understanding will help you not only solve problems but also verify whether your solution makes sense. Therefore, let's keep this fundamental principle in mind as we move forward to tackle the problem at hand.
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 is a simple notational change, but it helps in the subsequent algebraic manipulations. So, for our function f(x) = (1/9)x + 2, we replace f(x) with y, giving us the equation y = (1/9)x + 2. This step might seem trivial, but it's a crucial setup for the next steps. By using y, we are essentially representing the output of the function, which will become the input for the inverse function. This substitution makes it easier to visualize the relationship between the input and output variables, which is essential for understanding the concept of an inverse function. Furthermore, this change in notation allows us to treat the equation in a more standard algebraic form, making it easier to manipulate and solve for the inverse. Think of it as translating the function from a specific notation to a more general algebraic representation, which simplifies the process of finding the inverse. So, with this simple substitution, we are now ready to move on to the next step in our quest to find the inverse of the given function.
Step 2: Swap x and y
The next crucial step in finding the inverse of a function involves swapping the variables x and y. This is the heart of the process because it reflects the fundamental idea of an inverse function: reversing the roles of input and output. In our equation, y = (1/9)x + 2, we now interchange x and y, which gives us x = (1/9)y + 2. This swap is not just a mechanical step; it embodies the essence of inverting the function. By interchanging x and y, we are essentially saying that the output of the original function (y) now becomes the input of the inverse function, and the input of the original function (x) becomes the output of the inverse function. This step might seem abstract at first, but it's a powerful technique that allows us to algebraically manipulate the equation to solve for the inverse function. It's like looking at the function from a reversed perspective, where the roles of input and output are switched. This perspective shift is key to understanding and finding inverse functions. So, with this swap, we've set the stage for solving for y, which will ultimately give us the inverse function we're seeking.
Step 3: Solve for y
Now that we have swapped x and y, the next step is to isolate y on one side of the equation. This involves using algebraic manipulations to get y by itself. Starting with our equation x = (1/9)y + 2, we first want to get rid of the constant term on the right side. We can do this by subtracting 2 from both sides of the equation, which gives us x - 2 = (1/9)y. Next, to isolate y, we need to get rid of the fraction (1/9) multiplying y. We can do this by multiplying both sides of the equation by 9. This gives us 9(x - 2) = y. Now, we can simplify the left side by distributing the 9, resulting in 9x - 18 = y. We have now successfully isolated y. This step is crucial because it transforms the equation into a form where y is expressed in terms of x, which is exactly what we need for the inverse function. Solving for y is like unwrapping the original function, step by step, to reveal the inverse operation. Each algebraic manipulation we perform is a step towards isolating y and expressing it as a function of x. So, with y now isolated, we are just one step away from finding the inverse function.
Step 4: Replace y with f⁻¹(x)
The final step in finding the inverse of a function is to replace y with the inverse function notation, f⁻¹(x). This notation explicitly indicates that we have found the inverse of the original function f(x). In our case, we have y = 9x - 18. Replacing y with f⁻¹(x), we get f⁻¹(x) = 9x - 18. This is the inverse function of f(x) = (1/9)x + 2. This step is not just about changing notation; it's about formally declaring that we have successfully found the inverse function. The notation f⁻¹(x) is a standard way of representing the inverse, and it clearly communicates that this function undoes the operation of the original function f(x). Think of this step as putting the final piece of the puzzle in place. We've gone through all the algebraic manipulations, and now we're expressing the result in the correct notation. It's a formal way of saying, "We've found the inverse!" So, with this final notation change, we have completed the process of finding the inverse function, and we can confidently say that f⁻¹(x) = 9x - 18 is the inverse of f(x) = (1/9)x + 2.
Solution and Verification
Therefore, the inverse of the function f(x) = (1/9)x + 2 is h(x) = 9x - 18, which corresponds to option B. To verify this, we can compose the original function with its inverse and see if we get x. Let's calculate f(f⁻¹(x)):
f(f⁻¹(x)) = f(9x - 18) = (1/9)(9x - 18) + 2
Distributing the (1/9), we get:
f(f⁻¹(x)) = x - 2 + 2
Simplifying, we have:
f(f⁻¹(x)) = x
Similarly, let's calculate f⁻¹(f(x)):
f⁻¹(f(x)) = f⁻¹((1/9)x + 2) = 9((1/9)x + 2) - 18
Distributing the 9, we get:
f⁻¹(f(x)) = x + 18 - 18
Simplifying, we have:
f⁻¹(f(x)) = x
Since both f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, we have verified that h(x) = 9x - 18 is indeed the correct inverse function. This verification step is essential because it confirms that our calculations are correct and that the function we found truly undoes the original function. It's like a final check to ensure that our answer is accurate. By composing the function with its inverse and obtaining x, we have solid proof that we have found the correct inverse. So, with this verification, we can confidently conclude that option B is the correct answer.
Conclusion
In conclusion, finding the inverse of a function involves a series of steps that, when followed carefully, lead to the correct result. We started with the given function, f(x) = (1/9)x + 2, and systematically went through the process of replacing f(x) with y, swapping x and y, solving for y, and finally replacing y with f⁻¹(x). This process led us to the inverse function, f⁻¹(x) = 9x - 18. We then verified our result by composing the original function with its inverse, confirming that they indeed undo each other. Understanding the concept of inverse functions and the steps involved in finding them is a fundamental skill in algebra and calculus. It's not just about memorizing steps; it's about understanding the underlying principle of reversing the function's operation. This understanding allows you to apply the same techniques to a variety of functions and problems. So, whether you're solving equations, graphing functions, or exploring more advanced mathematical concepts, the ability to find the inverse of a function is a valuable tool in your mathematical toolkit. With practice and a solid understanding of the concepts, you can confidently tackle any inverse function problem that comes your way.
