Hey everyone! Today, we're diving into the exciting world of inverse equations. We've got a specific problem on our hands: finding the inverse of the equation (x-4)^2 - 2/3 = 6y - 12. This might seem a bit daunting at first, but don't worry, we'll break it down step by step and make sure you've got a solid understanding of the process. So, let's put on our math hats and get started!
Understanding Inverse Equations
Before we jump into solving this particular problem, let's quickly recap what inverse equations actually are. Think of it like this: an equation is a machine that takes an input (x) and spits out an output (y). An inverse equation is like running that machine in reverse – it takes the y value and gives you the original x value. In mathematical terms, to find the inverse of an equation, we essentially swap the roles of x and y and then solve for y. This process effectively "undoes" the original equation. Understanding this fundamental concept is crucial because it guides our entire approach to solving these types of problems. Without grasping the core idea of swapping variables and isolating y, we might get lost in the algebraic manipulations. So, keep this principle in mind as we move forward – it's the key to unlocking the solution.
When dealing with equations, especially those involving squares or other powers, it's super important to remember that there might be more than one solution. For example, when we take the square root to solve for a variable, we need to consider both the positive and negative roots. This is because both a positive number and its negative counterpart, when squared, will result in the same positive value. This concept is particularly relevant when we are finding inverse equations because swapping x and y can introduce these scenarios where multiple solutions arise. So, always keep an eye out for these situations and make sure you account for all possible solutions. Failing to do so can lead to incorrect answers and a misunderstanding of the full picture. Remember, mathematics is all about precision and completeness!
Also, remember that not all equations have inverses that are functions. An inverse function must pass the vertical line test, meaning that for each x value, there can only be one y value. If the original function doesn't pass the horizontal line test (meaning that a horizontal line intersects the graph at more than one point), its inverse will not be a function. This is a crucial concept in understanding the limitations and characteristics of inverse functions. When we encounter equations like the one we're tackling today, which involves a squared term, we need to be particularly mindful of this. The squared term often leads to a situation where the inverse is not a function unless we restrict the domain of the original function. Keep this in mind as we progress through the steps – it will help you anticipate potential challenges and interpret the final result more accurately.
Step-by-Step Solution
Okay, let's dive into the problem at hand. Our starting equation is: (x - 4)^2 - 2/3 = 6y - 12. Remember the golden rule for finding inverses? We need to swap x and y. So, let's do just that. Our equation now becomes: (y - 4)^2 - 2/3 = 6x - 12. See? We've simply interchanged the positions of x and y. This might seem like a small step, but it's the crucial foundation for finding the inverse. It's like switching the input and output labels on our equation machine. Now, our mission is to isolate y on one side of the equation. This will involve a series of algebraic manipulations, each designed to peel away the layers surrounding y. So, buckle up, and let's get to work!
Now, let's start isolating y. Our equation is (y - 4)^2 - 2/3 = 6x - 12. The first thing we want to do is get rid of that pesky -2/3. We can do this by adding 2/3 to both sides of the equation. This gives us: (y - 4)^2 = 6x - 12 + 2/3. Now, we need to simplify the right side. Let's combine the constants: -12 can be written as -36/3, so -36/3 + 2/3 = -34/3. Our equation now looks like this: (y - 4)^2 = 6x - 34/3. We're making progress! We've managed to isolate the squared term, which is a significant step forward. This means we're closer to getting y by itself. Remember, each step we take is like unwrapping a present, revealing more and more of the solution. So, let's keep going, and we'll have our answer in no time!
Next up, we need to get rid of that square. How do we do that? By taking the square root of both sides, of course! This gives us: y - 4 = ±√(6x - 34/3). Notice the ± sign? This is super important! Remember what we discussed earlier about square roots having two possible solutions – a positive and a negative one. We need to account for both of these possibilities to get the complete inverse. Ignoring this ± sign would be like only seeing half the picture, and we want the full, beautiful mathematical landscape. Now, we're almost there! We just have one more little step to take to completely isolate y.
Finally, to get y completely by itself, we need to add 4 to both sides of the equation. This gives us our final answer: y = 4 ± √(6x - 34/3). Ta-da! We've found the inverse equation. It might look a bit complex, but we got there step by step, and you followed along like a champ! This equation tells us that for a given x value, there are potentially two corresponding y values, which makes sense given the ± sign. This also highlights the fact that the inverse is not a function over its entire domain because it fails the vertical line test. But hey, we've successfully found the inverse equation, and that's a huge win!
Comparing with the Options
Now that we've found the inverse, let's take a look at the options provided and see which one matches our solution. We have:
A. y = (1/6)x^2 - (4/3)x + 43/9 B. y = 4 ± √(6x - 34/3) C. y = -4 ± √(6x - 34/3) D. -(x - 4)^2 - 2/3 = -6y + 12
Looking at these, it's clear that option B perfectly matches our solution: y = 4 ± √(6x - 34/3). Option A is a quadratic equation, which doesn't match the form of our inverse. Option C has a negative 4, which is incorrect. And option D is simply a manipulation of the original equation, not the inverse. So, we can confidently say that option B is the correct answer. You nailed it!
Key Takeaways
Alright, let's wrap things up by highlighting the key takeaways from this problem. First and foremost, remember the fundamental principle of finding inverse equations: swap x and y, and then solve for y. This is the golden rule that will guide you through any inverse equation problem. Secondly, always be mindful of the ± sign when taking square roots. It's crucial for capturing all possible solutions and understanding the complete picture. Finally, don't be afraid to break down the problem into smaller, manageable steps. Algebraic manipulations can seem intimidating, but by tackling them one at a time, you can conquer even the most complex equations. You've got this!
Practice Makes Perfect
Finding inverse equations might seem tricky at first, but like any skill, it gets easier with practice. So, don't stop here! Try tackling more problems, and you'll become a pro in no time. Maybe try varying the complexity of the equations you work with, or challenge yourself with problems that involve different types of functions, like exponential or logarithmic functions. The more you practice, the more confident you'll become in your ability to solve these problems. And remember, the journey of learning mathematics is all about continuous growth and improvement. So, keep pushing yourself, keep exploring, and keep having fun with it!
And hey, if you ever get stuck, don't hesitate to ask for help. There are tons of resources available, from online tutorials to textbooks to your friendly neighborhood math whiz. The key is to keep learning and keep growing. So, go out there and conquer those inverse equations! You've got this!
What is the inverse of the equation (x-4)^2 - 2/3 = 6y - 12?
Inverse Equation Explained Step-by-Step (x-4)^2 - 2/3 = 6y - 12
