Find The Inverse Of The Equation Y = 2x^2 - 8.

In the realm of mathematics, particularly in algebra and calculus, the concept of an inverse function is fundamental. An inverse function essentially reverses the operation of the original function. If a function ${ f(x) }$ takes an input ${ x }$ and produces an output ${ y }$, its inverse, denoted as ${ f^{-1}(x) }$, takes ${ y }$ as an input and returns the original ${ x }$. This article delves into the process of finding the inverse of a quadratic function, using the example ${ y = 2x^2 - 8 }$. Understanding how to find inverse functions is crucial for solving equations, simplifying expressions, and grasping more advanced mathematical concepts. The process involves a few key steps, including swapping the variables, isolating the dependent variable, and accounting for any domain restrictions that might arise due to the nature of the function. In our specific case, the quadratic nature of the function introduces some interesting challenges and considerations, especially concerning the range and domain of both the original function and its inverse. As we proceed, we will explore these aspects in detail, ensuring a comprehensive understanding of the topic. Mastery of this concept not only aids in solving mathematical problems but also enhances the ability to think critically and logically, skills that are valuable in various fields beyond mathematics.

The first crucial step in finding the inverse of a function is to swap the roles of ${ x }$ and ${ y }$. This reflects the fundamental idea of an inverse function: reversing the input and output. In our given equation, ${ y = 2x^2 - 8 }$, we replace every instance of ${ y }$ with ${ x }$ and every instance of ${ x }$ with ${ y }$. This transforms the original equation into ${ x = 2y^2 - 8 }$. This seemingly simple swap is the cornerstone of finding the inverse. By interchanging ${ x }$ and ${ y }$, we are essentially looking at the original function from a reversed perspective, setting the stage for expressing ${ y }$ in terms of ${ x }$, which is the very definition of an inverse function. This step is not just a mechanical procedure; it's a conceptual shift that allows us to see the relationship between the variables in reverse. It's important to perform this step accurately, as any error here will propagate through the rest of the process, leading to an incorrect inverse function. Once the variables are swapped, the next step involves algebraic manipulation to isolate ${ y }$, which will give us the equation for the inverse function. This initial swap is a critical gateway to unraveling the inverse relationship between the variables.

After swapping ${ x }$ and ${ y }$, our equation is now ${ x = 2y^2 - 8 }$. The next step involves isolating the term containing ${ y^2 }$. This is a standard algebraic manipulation technique used to solve for a particular variable. To isolate ${ 2y^2 }$, we first add 8 to both sides of the equation. This gives us ${ x + 8 = 2y^2 }$. The goal here is to peel away the layers surrounding the ${ y^2 }$ term, one step at a time. By adding 8 to both sides, we eliminate the constant term on the right side, bringing us closer to isolating the desired term. It's essential to maintain the balance of the equation, ensuring that any operation performed on one side is also performed on the other. This principle of algebraic manipulation is fundamental in solving equations. Once we have ${ x + 8 = 2y^2 }$, the next logical step is to eliminate the coefficient 2. We achieve this by dividing both sides of the equation by 2. This will further isolate ${ y^2 }$ and set the stage for the final step of solving for ${ y }$. Each of these steps is a deliberate effort to simplify the equation and bring us closer to expressing ${ y }$ in terms of ${ x }$.

Following the isolation of the ${ y^2 }$ term, our equation stands as ${ \frac{x + 8}{2} = y^2 }$. Now, to solve for ${ y }$, we need to take the square root of both sides of the equation. This operation is the key to unlocking ${ y }$ from its squared form. When taking the square root, it's crucial to remember that we must consider both the positive and negative roots. This is because both ${ \sqrt{A}^2 }$ and ${ (-\sqrt{A})^2 }$ equal ${ A }$. Therefore, taking the square root of both sides yields ( y = \pm \sqrt{\frac{x + 8}{2}} ). The ${ \pm }$ sign is a critical component of the solution. It signifies that for a given ${ x }$, there are two possible values of ${ y }$ that satisfy the equation, reflecting the quadratic nature of the original function. This is a characteristic feature of inverting quadratic functions, which, unlike linear functions, do not have a one-to-one correspondence between ${ x }$ and ${ y }$ over their entire domain. The inclusion of both the positive and negative roots is not just a mathematical formality; it's a reflection of the inherent symmetry of the quadratic function. Failing to include the ${ \pm }$ sign would result in an incomplete and incorrect inverse function. This step highlights the importance of careful consideration of all possible solutions when dealing with square roots in algebraic manipulations.

After solving for ${ y }$, we have ${ y = \pm \sqrt{\frac{x + 8}{2}} }$. This equation represents the inverse function of the original function ${ y = 2x^2 - 8 }$. The ${ \pm }$ sign indicates that the inverse is not a single function but rather two functions: ${ y = \sqrt{\frac{x + 8}{2}} }$ and ${ y = -\sqrt{\frac{x + 8}{2}} }$. This is a common characteristic of the inverses of quadratic functions. The inverse function effectively reverses the operation of the original function. If we input a value into the original function and then input the result into the inverse function, we should ideally get back our original input (with consideration for the domain restrictions, which we'll discuss later). In the context of the given options, this solution corresponds to option A, which is ${ y = \pm \sqrt{\frac{x + 8}{2}} }$. It's crucial to recognize that the inverse might not always be a function in the strictest sense, especially when dealing with functions that are not one-to-one. In such cases, the inverse might be a relation rather than a function. This is because a function requires each input to have a unique output, which might not be the case for the inverse of a non-one-to-one function. Understanding this distinction is vital in the broader study of functions and their inverses. Therefore, identifying the inverse function involves not only the algebraic manipulation but also the recognition of its nature and characteristics.

Now that we have derived the inverse function as ${ y = \pm \sqrt{\frac{x + 8}{2}} }$, let's analyze the given options to confirm our answer and understand why the other options are incorrect. Option A, ${ y = \pm \sqrt{\frac{x + 8}{2}} }$, perfectly matches our derived inverse function. This confirms that our algebraic steps were accurate and that we correctly identified the inverse. Moving on to option B, ${ y = \frac{\pm \sqrt{x + 8}}{2} }$, we can see a subtle but significant difference. The division by 2 is outside the square root, which is not what we derived. This discrepancy arises from an incorrect manipulation of the equation during the inversion process. The square root should encompass the entire fraction ${ \frac{x + 8}{2} }$, not just the ${ x + 8 }$ term. Option C, ${ y = \pm \sqrt{\frac{x}{2} + 8} }$, is also incorrect. Here, the 8 is added inside the square root after the division by 2, which is a deviation from the correct order of operations we followed when isolating ${ y }$. This error likely stems from a misunderstanding of how to properly reverse the operations performed in the original function. Finally, option D, ${ y = \frac{\pm \sqrt{x}}{2} + 4 }$, is the most different from our solution. This option seems to have incorrectly applied the inverse operations, leading to a completely different form of the equation. The addition of 4 outside the square root and the absence of the 8 within the square root are clear indicators that this is not the correct inverse. By systematically analyzing each option and comparing it to our derived solution, we can confidently confirm that option A is the correct inverse function. This exercise reinforces the importance of meticulous algebraic manipulation and a clear understanding of the steps involved in finding the inverse of a function.

In conclusion, after a step-by-step algebraic process, we have successfully determined that the inverse of the function ${ y = 2x^2 - 8 }$ is ${ y = \pm \sqrt{\frac{x + 8}{2}} }$. This corresponds to option A in the given choices. The process involved swapping the variables ${ x }$ and ${ y }$, isolating the ${ y^2 }$ term, and then taking the square root of both sides, remembering to include both the positive and negative roots. We also analyzed the other options and identified the errors in their forms, reinforcing our confidence in the correctness of option A. Finding the inverse of a function is a fundamental skill in mathematics, particularly in algebra and calculus. It requires a clear understanding of algebraic manipulations, careful attention to detail, and the ability to reverse the operations of the original function. The inverse of a quadratic function, like the one in this example, often introduces the concept of multiple solutions due to the ${ \pm }$ sign, highlighting the importance of considering both positive and negative roots when taking square roots. This exercise not only provides a solution to the specific problem but also enhances our understanding of inverse functions and the algebraic techniques used to find them. Mastery of these concepts is crucial for tackling more complex mathematical problems and for developing a deeper appreciation of the relationships between functions and their inverses.