Finding The Inverse Of Y = 2x² + 2 A Step-by-Step Guide

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In mathematics, the inverse of a function essentially "undoes" the operation performed by the original function. To find the inverse of a function, we swap the roles of the independent variable (x) and the dependent variable (y) and then solve for y. This process effectively reverses the mapping defined by the original function. When dealing with functions, especially those involving squares or other non-linear operations, it's crucial to remember that the inverse might not always be a function itself; it could be a relation. This means for a single input, there might be multiple outputs. In the context of this article, we will be meticulously walking through the steps to find the inverse of the given function y = 2x² + 2, paying close attention to algebraic manipulations and the implications of square roots. The goal is not just to arrive at the correct answer but also to understand the underlying principles of inverse functions and how they relate to the original functions. This exploration will provide a deeper insight into mathematical functions and their properties, enhancing problem-solving skills in algebra and beyond. Through a clear, step-by-step approach, we will ensure that the reader not only understands the mechanics of finding an inverse but also appreciates the mathematical concepts that underpin this process.

Step-by-Step Solution

To find the inverse of the function y = 2x² + 2, we will follow these steps:

  1. Swap x and y: This is the fundamental step in finding the inverse, where we interchange the roles of the independent and dependent variables. This reflects the idea that the inverse function reverses the mapping of the original function. By swapping x and y, we set up the equation in a form that allows us to solve for y in terms of x, which is the essence of finding an inverse function. This step is not merely a mechanical procedure but a conceptual shift that highlights the reversed relationship between the input and output of the original function. It's a critical step that sets the stage for the subsequent algebraic manipulations.

    • Starting equation: y = 2x² + 2
    • After swapping x and y: x = 2y² + 2

  2. Isolate the y² term: Our next goal is to isolate the term containing y². This involves performing algebraic operations to get the y² term by itself on one side of the equation. In our case, we first subtract 2 from both sides of the equation, and then we divide by 2. These steps are crucial in unraveling the operations performed on y in the original function. By isolating y², we prepare the equation for the final step of taking the square root, which will give us y in terms of x. This isolation process is a standard algebraic technique used in solving equations and is a key step in finding the inverse function.

    • Subtract 2 from both sides: x - 2 = 2y²
    • Divide both sides by 2: (x - 2) / 2 = y²

  3. Solve for y: To solve for y, we take the square root of both sides of the equation. It is critically important to remember that taking the square root introduces both positive and negative solutions. This is because both the positive and negative square roots, when squared, will yield the same positive number. This consideration is particularly important in the context of inverse functions, as it can lead to a multi-valued relation rather than a single-valued function. The inclusion of both positive and negative roots is a direct consequence of the properties of square roots and is a fundamental aspect of finding the inverse of functions involving squares.

    • Take the square root of both sides: y = ±√(( x - 2) / 2)
    • Simplify: y = ±√( x / 2 - 1)
    • Further simplification: y = ±√(1/2 * x - 1)

Identifying the Correct Option

By following the steps outlined above, we have found the inverse function to be y = ±√((1/2) * x - 1). Now, let's match our solution with the given options:

  • A. y = ±√( x - 2)
  • B. y = ±√(1/2 * x - 1)
  • C. y = ±√(2 * x² - 4)
  • D. y = ±√((1/2) * x - 1)

Comparing our derived inverse function with the options, we can clearly see that option D, y = ±√((1/2) * x - 1), matches our solution. Therefore, option D is the correct inverse of the function y = 2x² + 2. This methodical comparison ensures that we not only arrive at a solution but also verify its accuracy against the provided choices. The ability to match a derived result with given options is a crucial skill in mathematics, as it reinforces the understanding of the solution process and confirms the correctness of the final answer.

Detailed Explanation of Why Other Options Are Incorrect

Understanding why certain options are incorrect is as important as identifying the correct one. Let's delve into why options A, B, and C are not the inverses of the function y = 2x² + 2. This detailed analysis will provide a deeper understanding of the process of finding inverse functions and the common pitfalls to avoid.

  • Option A: y = ±√(x - 2)

    • This option seems plausible at first glance because it involves the square root and the x - 2 term, which appears in our solution process. However, the crucial difference lies in the absence of the division by 2 inside the square root. If we were to substitute this inverse back into the original function, we would not obtain the identity x, which is a necessary condition for inverse functions. The algebraic steps required to isolate y² demand that we divide by 2, and the omission of this division in option A makes it an incorrect choice. This highlights the importance of meticulously following each step in the process of finding an inverse, as even seemingly minor discrepancies can lead to an incorrect result.

  • Option B: y = ±√(1/2 * x - 1)

    • This option is very close to the correct answer. In fact, it is the correct inverse function. So, we will skip the detailed explanation here and move on to the next option.

  • Option C: y = ±√(2 * x² - 4)

    • This option is structurally very different from our derived solution. The presence of the x² term inside the square root immediately indicates that it cannot be the correct inverse. When finding the inverse, we manipulate the original equation to express y in terms of x. The inverse function should undo the operations of the original function, and the x² term in this option suggests an entirely different set of operations. Furthermore, the term -4 under the square root is inconsistent with the algebraic manipulations we performed. This option serves as a good example of how a superficial similarity to the original function can be misleading, and a thorough understanding of the inverse function process is crucial for avoiding such errors.

By dissecting these incorrect options, we reinforce our understanding of the correct methodology for finding inverse functions and learn to recognize common errors. This critical analysis is an invaluable part of the learning process, as it strengthens our ability to apply mathematical concepts accurately and confidently.

Conclusion

In summary, to find the inverse of the function y = 2x² + 2, we followed the standard procedure of swapping x and y, isolating the y² term, and solving for y. This process led us to the inverse function y = ±√((1/2) * x - 1), which corresponds to option D. We also examined why the other options were incorrect, emphasizing the importance of meticulous algebraic manipulation and a thorough understanding of the properties of inverse functions. This exercise not only provides a solution to the specific problem but also enhances our overall problem-solving skills in mathematics. The ability to find the inverse of a function is a fundamental concept in algebra and calculus, with applications in various fields, including physics, engineering, and computer science. By mastering this skill, we equip ourselves with a powerful tool for understanding and manipulating mathematical relationships.