Finding The Inverse Of F(x) = (1/9)x + 2 A Step By Step Guide

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Determining the inverse of a function is a fundamental concept in mathematics, particularly in algebra and calculus. The inverse function, denoted as f⁻¹(x), essentially reverses the operation of the original function, f(x). In simpler terms, if f(a) = b, then f⁻¹(b) = a. This article will delve into the process of finding the inverse of the linear function f(x) = (1/9)x + 2, offering a comprehensive explanation and a step-by-step approach to solve this type of problem. Understanding inverse functions is crucial for various mathematical applications, including solving equations, analyzing function behavior, and exploring the relationship between different functions. Let's embark on this journey to unravel the intricacies of inverse functions and master the technique of finding them. This understanding will not only help in solving specific problems but also build a solid foundation for more advanced mathematical concepts.

Understanding Inverse Functions

Before diving into the specific function f(x) = (1/9)x + 2, it's crucial to grasp the core concept of inverse functions. An inverse function, in essence, undoes what the original function does. Imagine a function as a machine that takes an input, processes it, and produces an output. The inverse function is like a machine that takes that output and returns the original input. Mathematically, if we have a function f that maps x to y, then its inverse, denoted as f⁻¹*, maps y back to x. This relationship can be expressed as f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. Not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each input corresponds to a unique output, and each output corresponds to a unique input. Graphically, a one-to-one function passes the horizontal line test: any horizontal line drawn across the graph intersects the function at most once. Linear functions, except for horizontal lines, are always one-to-one and thus have inverses. The process of finding an inverse involves swapping the roles of the input and output variables and then solving for the new output variable. This technique effectively reverses the operations performed by the original function. Mastering the concept of inverse functions opens doors to a deeper understanding of mathematical relationships and problem-solving strategies. It allows us to see functions not just as operations but as reversible transformations, enriching our mathematical intuition.

Step-by-Step Method to Find the Inverse Function

To find the inverse of the function f(x) = (1/9)x + 2, we'll follow a systematic, step-by-step approach. This method is applicable to a wide range of functions and provides a clear roadmap for solving inverse function problems. Each step is crucial, and understanding the reasoning behind each step is as important as the execution itself.

Step 1: Replace f(x) with y

The first step in finding the inverse is to replace the function notation f(x) with the variable y. This substitution simplifies the equation and makes it easier to manipulate algebraically. So, f(x) = (1/9)x + 2 becomes y = (1/9)x + 2. This step is a simple change in notation but sets the stage for the subsequent algebraic manipulations. It's like rewriting a sentence in a more convenient format for analysis.

Step 2: Swap x and y

The core concept of finding an inverse is to reverse the roles of the input and output. This is achieved by swapping the variables x and y in the equation. So, y = (1/9)x + 2 becomes x = (1/9)y + 2. This step directly reflects the idea that the inverse function undoes the original function. By swapping x and y, we are essentially asking, "What input y would produce the output x in the original function?"

Step 3: Solve for y

Now, the goal is to isolate y on one side of the equation. This involves performing algebraic operations to undo the operations acting on y. In the equation x = (1/9)y + 2, we first subtract 2 from both sides to get x - 2 = (1/9)y. Then, to get y by itself, we multiply both sides of the equation by 9, resulting in 9(x - 2) = y. This step requires careful application of algebraic principles to ensure that the equation remains balanced. The order of operations must be reversed to isolate y correctly.

Step 4: Simplify and Express as f⁻¹(x)

After solving for y, we simplify the expression and replace y with the inverse function notation, f⁻¹(x). Expanding 9(x - 2) gives 9x - 18. Therefore, y = 9x - 18. Finally, we express the inverse function as f⁻¹(x) = 9x - 18. This step completes the process of finding the inverse and presents the result in standard function notation. It's the final polish that makes the solution clear and understandable.

Applying the Steps to f(x) = (1/9)x + 2

Let's apply the steps outlined above to find the inverse of the function f(x) = (1/9)x + 2. This will provide a concrete example of how the method works in practice and solidify your understanding.

Step 1: Replace f(x) with y

Replace f(x) with y: y = (1/9)x + 2

This initial step transforms the function notation into a more workable equation. It's a simple substitution, but it's a necessary prelude to the next steps.

Step 2: Swap x and y

Swap x and y: x = (1/9)y + 2

This is the crucial step where we reverse the roles of input and output. By swapping x and y, we are setting up the equation to solve for the inverse function.

Step 3: Solve for y

  1. Subtract 2 from both sides: x - 2 = (1/9)y
  2. Multiply both sides by 9: 9(x - 2) = y

These algebraic manipulations isolate y on one side of the equation. Each operation is performed carefully to maintain the equality and move closer to the solution.

Step 4: Simplify and Express as f⁻¹(x)

  1. Simplify: y = 9x - 18
  2. Express as inverse function: f⁻¹(x) = 9x - 18

This final step presents the inverse function in standard notation. We have successfully found the inverse of f(x) = (1/9)x + 2, which is f⁻¹(x) = 9x - 18.

Verification of the Inverse Function

To ensure that f⁻¹(x) = 9x - 18 is indeed the inverse of f(x) = (1/9)x + 2, we can verify it using the property that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. This verification step is crucial for confirming the correctness of the solution and provides a deeper understanding of the relationship between a function and its inverse.

Verification using f⁻¹(f(x)) = x

  1. Substitute f(x) into f⁻¹(x): f⁻¹(f(x)) = 9[(1/9)x + 2] - 18
  2. Distribute: 9[(1/9)x + 2] - 18 = x + 18 - 18
  3. Simplify: x + 18 - 18 = x

The result, x, confirms that f⁻¹(f(x)) = x, which is one part of the verification process.

Verification using f(f⁻¹(x)) = x

  1. Substitute f⁻¹(x) into f(x): f(f⁻¹(x)) = (1/9)(9x - 18) + 2
  2. Distribute: (1/9)(9x - 18) + 2 = x - 2 + 2
  3. Simplify: x - 2 + 2 = x

The result, x, confirms that f(f⁻¹(x)) = x, which completes the verification process.

Since both conditions, f⁻¹(f(x)) = x and f(f⁻¹(x)) = x, are satisfied, we can confidently conclude that f⁻¹(x) = 9x - 18 is indeed the inverse of f(x) = (1/9)x + 2. This verification step not only confirms the solution but also reinforces the fundamental concept of inverse functions.

Conclusion

In conclusion, the inverse of the function f(x) = (1/9)x + 2 is f⁻¹(x) = 9x - 18. We arrived at this solution by following a clear, step-by-step method: replacing f(x) with y, swapping x and y, solving for y, and expressing the result as f⁻¹(x). Furthermore, we verified our solution by confirming that both f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. This process not only provides the answer but also deepens the understanding of inverse functions and their properties. The ability to find and verify inverse functions is a valuable skill in mathematics, with applications ranging from solving equations to understanding function transformations. By mastering this technique, you enhance your mathematical toolkit and gain a more profound appreciation for the interconnectedness of mathematical concepts. The journey of finding the inverse function is not just about arriving at the answer; it's about understanding the process and the underlying principles that make it work. This understanding empowers you to tackle a wider range of mathematical problems with confidence and clarity.

The correct answer is B. h(x) = 9x - 18.