Finding The Inverse Of $(x-4)^2-2/3=6y-12$ Equation A Step-by-Step Guide
Understanding Inverse Equations
In mathematics, finding the inverse of an equation is a fundamental concept that allows us to "undo" the original function. An inverse equation essentially reverses the roles of the input (x) and the output (y). This process is crucial in various mathematical applications, from solving equations to understanding the relationships between different functions. In this article, we will delve into the step-by-step method of finding the inverse of the equation (x-4)^2- rac{2}{3}=6 y-12. We will explore each step in detail, ensuring a clear understanding of the underlying principles and techniques. This comprehensive guide aims to equip you with the knowledge and skills necessary to confidently tackle similar problems. Before we dive into the specifics of the given equation, let's first clarify what an inverse equation means and why it is significant.
An inverse equation essentially reverses the roles of the input (x) and the output (y). If a function takes an input x and produces an output y, its inverse function takes y as input and produces x as output. This reversal is fundamental to understanding the relationship between a function and its inverse. Graphically, the inverse function is a reflection of the original function across the line y = x. This visual representation helps in understanding the symmetry and the reversed relationship between the two functions. The concept of inverse functions is not just a theoretical construct; it has practical applications in various fields. For instance, in cryptography, inverse functions are used to decode encrypted messages. In computer graphics, they are used to transform objects back to their original positions. In calculus, inverse functions are essential for integration and differentiation. Therefore, mastering the techniques to find inverse functions is crucial for a strong foundation in mathematics.
Step 1: Swap x and y
The first and most crucial step in finding the inverse of an equation is to swap the variables x and y. This step embodies the fundamental concept of inverting a function, where the roles of input and output are reversed. By interchanging x and y, we set the stage for rearranging the equation to solve for y, which will then represent the inverse function. This seemingly simple step is the cornerstone of the entire process, and a clear understanding of its significance is vital. Let's illustrate this with our given equation: (x-4)^2- rac{2}{3}=6 y-12. By swapping x and y, we obtain (y-4)^2- rac{2}{3}=6 x-12. This new equation now represents the inverse relationship, but it is not yet in the standard form of y as a function of x. The subsequent steps will focus on isolating y on one side of the equation, thereby expressing the inverse function explicitly. This initial swap is not just a mechanical step; it is a conceptual shift that reflects the reversed relationship between the original function and its inverse. It is the foundation upon which the rest of the solution is built, and a thorough grasp of this step is essential for successfully finding inverse equations.
Step 2: Isolate the Term with y
After swapping x and y, our next goal is to isolate the term containing y. This involves performing algebraic operations to move all other terms to the opposite side of the equation. The aim is to get the term with y by itself, making it easier to eventually solve for y. This process often requires adding, subtracting, multiplying, or dividing terms on both sides of the equation, ensuring that the equation remains balanced. In our specific case, after swapping x and y, we have the equation (y-4)^2- rac{2}{3}=6 x-12. To isolate the term with y, which is , we need to eliminate the - rac{2}{3} term on the left side. We can do this by adding rac{2}{3} to both sides of the equation. This gives us (y-4)^2 = 6x - 12 + rac{2}{3}. Simplifying the right side, we combine the constant terms: -12 + rac{2}{3} = - rac{36}{3} + rac{2}{3} = - rac{34}{3}. Thus, our equation becomes (y-4)^2 = 6x - rac{34}{3}. This step is crucial because it brings us closer to isolating y. By isolating the term with y, we set the stage for the next step, which involves undoing the operations applied to y, such as squaring or taking a root. This methodical isolation process is a common technique in algebra and is essential for solving various types of equations.
Step 3: Take the Square Root
Having isolated the term with y squared, the next logical step is to take the square root of both sides of the equation. This operation helps us to eliminate the square and bring us closer to isolating y. However, it is crucial 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 original value. Therefore, we must consider both possibilities to ensure we capture the complete inverse relationship. In our case, we have the equation (y-4)^2 = 6x - rac{34}{3}. Taking the square root of both sides, we get \sqrt{(y-4)^2} = \pm \sqrt{6x - rac{34}{3}}. This simplifies to y-4 = \pm \sqrt{6x - rac{34}{3}}. The symbol is essential here, as it indicates that there are two possible solutions for y. Neglecting this symbol would lead to an incomplete solution, missing one branch of the inverse function. This step highlights the importance of careful consideration of mathematical principles when manipulating equations. Taking the square root is a common operation, but it must be done with the awareness of its implications, particularly the introduction of both positive and negative roots. The next step will involve isolating y completely by addressing the remaining term on the left side of the equation.
Step 4: Isolate y
The final step in finding the inverse is to isolate y completely. This involves performing any remaining algebraic operations to get y by itself on one side of the equation. In our case, we have the equation y-4 = \pm \sqrt{6x - rac{34}{3}}. To isolate y, we need to eliminate the ā-4ā term on the left side. This can be achieved by adding 4 to both sides of the equation. Adding 4 to both sides, we get y = 4 \pm \sqrt{6x - rac{34}{3}}. This equation now represents the inverse of the original function. It expresses y explicitly in terms of x, and it captures the dual nature of the inverse due to the sign. This final step is the culmination of all the previous steps, bringing us to the solution we sought. By systematically swapping variables, isolating terms, and undoing operations, we have successfully found the inverse equation. This process not only provides the solution but also reinforces the importance of methodical algebraic manipulation. The resulting equation, y = 4 \pm \sqrt{6x - rac{34}{3}}, is the inverse of the original equation (x-4)^2- rac{2}{3}=6 y-12. It is crucial to understand that this inverse represents a relation, not necessarily a function, due to the sign. This means that for a single value of x, there may be two corresponding values of y. This distinction is important in understanding the properties and behavior of inverse relations and functions.
Conclusion
In summary, finding the inverse of the equation (x-4)^2- rac{2}{3}=6 y-12 involves a series of methodical steps. First, we swapped x and y, which is the fundamental step in finding an inverse. Then, we isolated the term containing y, followed by taking the square root of both sides, remembering to include both positive and negative roots. Finally, we isolated y completely to arrive at the inverse equation: y = 4 \pm \sqrt{6x - rac{34}{3}}. This process highlights the importance of careful algebraic manipulation and a thorough understanding of mathematical principles. The final answer, y = 4 \pm \sqrt{6x - rac{34}{3}}, corresponds to option B in the given choices. This exercise demonstrates a systematic approach to finding inverse equations, a crucial skill in mathematics with applications in various fields. Understanding the steps involved and the reasoning behind them is key to successfully tackling similar problems. The process of finding inverses not only enhances algebraic skills but also deepens the understanding of the relationship between functions and their inverses. The ability to find inverses is a valuable tool in mathematical problem-solving and is essential for further studies in mathematics and related disciplines.
Therefore, the correct answer is B. y=4 rac{}{\pm} \sqrt{6 x-\frac{34}{3}}.
