Finding The Inverse Of F(x) = 9x + 7 A Step-by-Step Solution

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In the realm of mathematics, inverse functions play a crucial role in undoing the operations performed by the original function. Understanding how to find the inverse of a function is a fundamental skill, especially when dealing with linear functions. This article delves into the process of finding the inverse of the function f(x) = 9x + 7, providing a step-by-step guide and a detailed explanation to ensure clarity and comprehension. Our journey will cover the essential concepts, the methodical approach, and the correct solution, empowering you to confidently tackle similar problems.

Understanding Inverse Functions

Before we embark on the solution, let's first grasp the essence of inverse functions. An inverse function, denoted as f⁻¹(x), essentially reverses the operation of the original function, f(x). If f(a) = b, then f⁻¹(b) = a. In simpler terms, if you input 'a' into the function f, and the output is 'b', then inputting 'b' into the inverse function f⁻¹ will give you 'a'. This reversal property is the core of inverse functions.

Key Characteristics of Inverse Functions

  • One-to-one correspondence: For a function to have an inverse, it must be one-to-one. This means that each input value (x) corresponds to a unique output value (y), and vice versa. In graphical terms, a one-to-one function passes both the horizontal and vertical line tests.
  • Domain and Range Swap: The domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse function. This swap is a direct consequence of the reversal property.
  • Reflection Across y = x: The graphs of a function and its inverse are reflections of each other across the line y = x. This visual representation provides a clear understanding of the inverse relationship.

Why are Inverse Functions Important?

Inverse functions have a wide range of applications in various fields, including:

  • Solving Equations: They are essential tools for solving equations, especially when the variable is trapped inside a function.
  • Cryptography: Inverse functions play a crucial role in encryption and decryption processes.
  • Calculus: In calculus, inverse functions are used in differentiation and integration.
  • Real-world Applications: Inverse functions are used in diverse applications such as converting units, calculating discounts, and modeling inverse relationships in physics and engineering.

Step-by-Step Guide to Finding the Inverse of f(x) = 9x + 7

Now, let's dive into the process of finding the inverse of the given function, f(x) = 9x + 7. We will follow a systematic approach to ensure accuracy and clarity.

Step 1: Replace f(x) with y

This step is a simple notational change that makes the subsequent steps easier to follow. Replacing f(x) with y, we get:

y = 9x + 7

This equation represents the same relationship as the original function, but in a form that is more convenient for the next steps.

Step 2: Swap x and y

This is the crucial step where we initiate the reversal process. By swapping x and y, we are essentially interchanging the input and output variables, which is the core concept of finding an inverse function. Swapping x and y in the equation y = 9x + 7, we get:

x = 9y + 7

This new equation represents the inverse relationship, but it is not yet in the standard form of a function, where y is expressed in terms of x.

Step 3: Solve for y

Our goal now is to isolate y on one side of the equation. This involves performing algebraic manipulations to undo the operations that are being applied to y. In this case, we need to subtract 7 from both sides and then divide by 9.

  1. Subtract 7 from both sides:

    x - 7 = 9y + 7 - 7 x - 7 = 9y

  2. Divide both sides by 9:

    (x - 7) / 9 = 9y / 9 (x - 7) / 9 = y

Now we have y isolated on one side of the equation:

y = (x - 7) / 9

Step 4: Replace y with f⁻¹(x)

Finally, we replace y with the notation for the inverse function, f⁻¹(x). This is a symbolic representation that indicates we have found the inverse function.

f⁻¹(x) = (x - 7) / 9

This is the inverse function of f(x) = 9x + 7. It expresses the inverse relationship in the standard function notation.

Analyzing the Solution

We have successfully found the inverse function, f⁻¹(x) = (x - 7) / 9. To ensure a complete understanding, let's analyze the solution and verify its correctness.

Rewriting the Inverse Function

We can rewrite the inverse function to match the format of the multiple-choice options. Distributing the division by 9, we get:

f⁻¹(x) = (x / 9) - (7 / 9)

f⁻¹(x) = (1/9)x - (7/9)

This form of the inverse function clearly matches option A in the given choices.

Verification of the Inverse Function

To verify that we have indeed found the correct inverse function, we can use the property that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Let's test this property:

  1. f(f⁻¹(x)) = f((1/9)x - (7/9)) = 9((1/9)x - (7/9)) + 7 = x - 7 + 7 = x

  2. f⁻¹(f(x)) = f⁻¹(9x + 7) = ((9x + 7) - 7) / 9 = (9x) / 9 = x

Both compositions result in x, confirming that f⁻¹(x) = (1/9)x - (7/9) is indeed the inverse function of f(x) = 9x + 7.

Common Mistakes to Avoid

Finding inverse functions involves a few steps where errors can easily occur. Here are some common mistakes to watch out for:

  • Incorrectly Swapping x and y: Swapping x and y is a crucial step. Ensure you swap the variables correctly, otherwise, the entire process will lead to an incorrect result.
  • Algebraic Errors in Solving for y: Solving for y involves algebraic manipulations. Be careful with signs, order of operations, and distribution. A small error in algebra can lead to a wrong inverse function.
  • Forgetting to Replace y with f⁻¹(x): The final step is to replace y with the notation f⁻¹(x). Forgetting this step means you haven't expressed the answer in the correct notation for an inverse function.
  • Not Verifying the Solution: It's always a good practice to verify your solution by checking if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This helps catch any errors and ensures you have the correct inverse function.

Conclusion

In this comprehensive guide, we have meticulously explored the process of finding the inverse of the function f(x) = 9x + 7. We began by understanding the fundamental concepts of inverse functions, including their key characteristics and importance. We then followed a step-by-step approach, replacing f(x) with y, swapping x and y, solving for y, and finally, replacing y with f⁻¹(x). We analyzed the solution, verified its correctness, and discussed common mistakes to avoid. By understanding and mastering this process, you can confidently find the inverses of linear functions and apply this knowledge to various mathematical problems and real-world scenarios. The correct answer is A. f⁻¹(x) = (1/9)x - (7/9). Remember, practice makes perfect, so continue to solve similar problems to solidify your understanding of inverse functions.