Finding The Inverse Of Y=7x^2-10 A Step-by-Step Solution
In mathematics, finding the inverse of a function essentially means reversing the roles of the input (x) and output (y). This process can be particularly interesting when dealing with quadratic functions, which have a distinctive U-shaped curve. In this comprehensive guide, we will delve into the method of finding the inverse of a quadratic function, using the example y = 7x² - 10. We will break down each step, clarify the underlying principles, and also discuss how to handle the complexities that arise due to the nature of quadratic functions. Understanding inverses is crucial in various mathematical applications, and this guide aims to provide a clear and thorough understanding of the process. The process of finding the inverse of a function involves a few key steps. First, we replace f(x) with y. Then, we swap x and y in the equation. After swapping the variables, we solve the equation for y. This new equation represents the inverse function, often denoted as f⁻¹(x). However, quadratic functions introduce an additional layer of complexity because their parabolic shape means that they don't always have a one-to-one correspondence between x and y values. This can lead to some interesting considerations when determining the inverse. For the given function, y = 7x² - 10, we follow these steps carefully to determine its inverse. By understanding this process, we can appreciate the nature of inverse functions and their importance in mathematical analysis. The following sections will provide a detailed walkthrough of this process.
Step 1: Swap x and y
The first key step in finding the inverse of any function is to swap the variables x and y. This might seem like a simple step, but it is the foundational element that allows us to reverse the roles of the input and output. In essence, what was previously the dependent variable (y) now becomes the independent variable, and vice versa. This swapping action sets the stage for solving the equation in terms of the new y, which will represent the inverse function. For our specific function, y = 7x² - 10, swapping x and y gives us x = 7y² - 10. This new equation now expresses x in terms of y, which is the critical transformation needed to find the inverse. We are now poised to manipulate this equation algebraically to isolate y on one side, thus defining the inverse function. This step is crucial because it reflects the fundamental concept of an inverse function – that it undoes what the original function does. By switching x and y, we are essentially looking at the function from the perspective of reversing its operation. This technique is universally applicable across various types of functions, but it is particularly insightful in the context of quadratic functions, where the symmetry of the parabola introduces additional considerations. In the subsequent steps, we will solve this equation for y, revealing the algebraic form of the inverse. Understanding the significance of this initial swap is vital for mastering the process of inverse function determination.
Step 2: Isolate the y² Term
After swapping x and y, our next goal is to isolate the y² term in the equation. This is a crucial step in solving for y and ultimately finding the inverse function. Isolating the y² term means getting it by itself on one side of the equation, which prepares us to take the square root and solve for y. For the equation x = 7y² - 10, the first step in isolating 7y² is to add 10 to both sides. This gives us x + 10 = 7y². Now, we have moved the constant term to the same side as x, bringing us closer to isolating the squared term. Next, we need to get rid of the coefficient 7 that is multiplying y². To do this, we divide both sides of the equation by 7, resulting in (x + 10)/7 = y². This step completely isolates the y² term, making it clear that y² is equal to the expression (x + 10)/7. We are now in a position to take the square root of both sides, which will be the next step in solving for y. Isolating the y² term is not only a necessary algebraic manipulation but also a logical step in reversing the operations of the original function. In the original function, we first squared x, then multiplied by 7, and finally subtracted 10. To find the inverse, we are undoing these operations in reverse order. Thus, by adding 10 and then dividing by 7, we are effectively reversing the last two operations performed by the original function. This process highlights the inverse relationship between the original function and its inverse.
Step 3: Take the Square Root
With the y² term isolated, the next crucial step is to take the square root of both sides of the equation. This operation is key to solving for y and obtaining the expression for the inverse function. When taking the square root, it's essential to remember that we need to consider both the positive and negative roots. This is because both the positive and negative square roots, when squared, will yield the same positive value. For our equation, (x + 10)/7 = y², taking the square root of both sides gives us y = ±√((x + 10)/7). The ± sign is critical here. It indicates that for a given value of x, there are two possible values of y that could satisfy the equation. This is a direct consequence of the fact that quadratic functions produce the same y value for both a positive and a negative x value (due to the squaring operation). The inclusion of both positive and negative roots reflects the symmetry inherent in the parabolic shape of quadratic functions. By considering both roots, we acknowledge that the inverse relation is not a function in the traditional sense unless we restrict the domain of the original function. This step underscores a significant characteristic of inverse functions of quadratics: they often require careful consideration of the domain and range to ensure they are properly defined. The ± sign is not just a mathematical formality; it is a reflection of the fundamental symmetry of the quadratic function and the need to account for both possibilities when reversing the function's operation. Now, we have an expression for y that involves x, and this expression represents the inverse relation of our original function.
The Inverse Function
After completing the algebraic steps, we have arrived at the inverse function. The inverse function, in this case, is represented by the equation y = ±√((x + 10)/7). This equation gives us the values of y that correspond to x in the inverse relation. However, it's important to note that this relation is not a function in the strictest sense, because for a single value of x, there are two possible values of y (one positive and one negative), due to the ± sign. This situation arises because the original quadratic function, y = 7x² - 10, is not one-to-one. A one-to-one function is one where each y value corresponds to only one x value. Quadratic functions fail this test because the parabolic shape means that each y value (except for the vertex) corresponds to two x values. To make the inverse a true function, we would need to restrict the domain of the original function. For example, we could consider only the part of the parabola where x ≥ 0 or only the part where x ≤ 0. This restriction would eliminate the symmetry that leads to the ± sign in the inverse. The inverse function, y = ±√((x + 10)/7), reveals how the operations of the original function have been reversed. The original function squares x, multiplies by 7, and then subtracts 10. The inverse function adds 10 to x, divides by 7, and then takes the square root (both positive and negative). This reversal of operations is the essence of an inverse function. In the context of multiple-choice questions, such as the one posed, the correct option would be the one that matches this derived inverse function. Understanding the steps and the underlying principles allows us to confidently identify the correct answer.
Choosing the Correct Option
Having derived the inverse function as y = ±√((x + 10)/7), we can now confidently choose the correct option from the given alternatives. Let's revisit the options presented:
A. y = (±√(x + 10))/7 B. y = ±√((x + 10)/7) C. y = ±√(x/7 + 10) D. y = (±√(x))/7 ± (√10)/7
By comparing our derived inverse function with the options, it is clear that option B, y = ±√((x + 10)/7), exactly matches our result. Therefore, option B is the correct answer. The other options can be eliminated because they do not correctly represent the inverse function we derived. Option A incorrectly divides the entire square root expression by 7, while the 7 should be inside the square root as the denominator. Option C incorrectly adds 10 inside the square root after dividing x by 7, instead of adding 10 to x before dividing by 7. Option D is a more complex expression that does not follow the correct order of operations for reversing the original function. This process of elimination, based on our understanding of the steps to find the inverse and the correct algebraic form, is a powerful tool in problem-solving. It allows us to not only identify the correct answer but also to understand why the other options are incorrect. This deeper understanding is valuable for tackling similar problems in the future. Choosing the correct option is not just about matching the answer; it's about verifying that the answer aligns with the mathematical principles and steps we have followed. In this case, option B is the only one that accurately reflects the inverse of the given quadratic function.
Conclusion
In conclusion, finding the inverse of the quadratic function y = 7x² - 10 involves several key steps: swapping x and y, isolating the y² term, and taking the square root of both sides. This process leads us to the inverse relation y = ±√((x + 10)/7). The ± sign is a critical aspect of this result, highlighting that the inverse is not a function in the traditional sense unless we restrict the domain of the original function. Understanding the steps and the underlying principles is essential for correctly determining the inverse of a function. The correct answer from the given options is B, y = ±√((x + 10)/7), which accurately represents the inverse we derived. This exercise demonstrates the importance of careful algebraic manipulation and a thorough understanding of the nature of inverse functions, particularly in the context of quadratic functions. The symmetry of quadratic functions and the implications for their inverses are key concepts that every math student should grasp. By mastering this process, one can confidently tackle similar problems and gain a deeper appreciation for the relationships between functions and their inverses. The ability to find the inverse of a function is a fundamental skill in mathematics, with applications in various areas such as calculus, algebra, and mathematical analysis. It is also a key concept in understanding transformations and the reversibility of mathematical operations.
