Inverse Of Y=5x^2+10: Find The Simplified Equation

by Mei Lin 51 views

Hey guys! Today, we're diving into the fascinating world of inverse functions, specifically how to find the inverse of the equation $y=5x^2+10$. This might sound intimidating, but don't worry, we'll break it down step by step. Let's get started!

Understanding Inverse Functions

Before we jump into the equation, let's quickly recap what inverse functions are all about. Think of a function as a machine: you put something in (the input, usually x), and the machine spits out something else (the output, usually y). An inverse function is like a machine that does the exact opposite – it takes the output (y) and spits out the original input (x). Essentially, it undoes what the original function did. This undoing process is crucial in many areas of mathematics and has practical applications in various fields.

To find the inverse, we essentially swap the roles of x and y and then solve for y. This might sound simple, but it's a fundamental concept that helps us understand how functions relate to each other. Understanding inverse functions isn't just about manipulating equations; it's about understanding the relationship between inputs and outputs and how mathematical operations can be reversed. This concept is essential in various mathematical fields, such as calculus and differential equations, where understanding the inverse of a transformation can significantly simplify complex problems. Additionally, the idea of reversing a process is crucial in cryptography, where encoding and decoding messages rely on inverse operations. So, grasping this concept not only enhances your mathematical toolkit but also opens doors to understanding more advanced and real-world applications.

The Key Step: Swapping x and y

So, the very first and most important step in finding the inverse of any function is to swap x and y. This is the magic trick that sets everything in motion! In our case, we start with the equation $y=5x^2+10$. Swapping x and y gives us: $x=5y^2+10$. This new equation represents the inverse relationship. We've essentially rewritten the equation to express x in terms of y, which is the first part of finding the inverse function. This simple swap is the foundation upon which we'll build the rest of the solution. It's like flipping a coin; it changes the perspective and sets us on the path to the inverse. Remember, the inverse function does the opposite of the original, so swapping the input and output variables is the logical first step.

It might seem like a small change, but it's a crucial one. Think of it like this: if the original function takes an x value and squares it, multiplies by 5, and then adds 10 to get y, the inverse function needs to start with that y value and undo those operations in reverse order to get back to the original x. Swapping x and y sets up the equation so we can isolate y and find out exactly what those reverse operations are. So, when you're faced with finding an inverse, remember this crucial swap – it's the key to unlocking the solution!

Isolating y: Solving for the Inverse

Now that we've swapped x and y, we have the equation $x=5y^2+10$. Our next mission, should we choose to accept it, is to isolate y. This means getting y all by itself on one side of the equation. To do this, we need to carefully undo the operations that are being performed on y, following the reverse order of operations (PEMDAS/BODMAS in reverse!).

First, we need to get rid of that pesky +10. We can do this by subtracting 10 from both sides of the equation:

x−10=5y2x - 10 = 5y^2

Next up, we need to deal with the multiplication by 5. To undo this, we divide both sides of the equation by 5:

x−105=y2\frac{x-10}{5} = y^2

Now we're getting closer! We have y squared, but we want just y. To undo the squaring, we take the square root of both sides:

±x−105=y\pm\sqrt{\frac{x-10}{5}} = y

Notice the ± sign? This is super important! When we take the square root, we need to consider both the positive and negative roots. This is because both a positive and a negative number, when squared, will result in a positive number. For example, both 2² and (-2)² equal 4. So, we need to account for both possibilities when finding the inverse.

Identifying the Correct Simplified Equation

Looking back at the original question, we were given a few options for the simplified equation that represents the inverse. Based on our step-by-step solution, the equation $x=5y^2+10$ is the one that can be simplified to find the inverse of $y=5x^2+10$. We've successfully swapped x and y, which is the first crucial step in finding the inverse function. While we went on to solve for y completely, the question specifically asked which equation can be simplified to find the inverse, and that's precisely what $x=5y^2+10$ does. It sets the stage for the rest of the process.

So, the other options – $\frac{1}{y}=5x^2+10$, $-y=5x^2+10$, and $y=\frac{1}{5}x^2+\frac{1}{10}$ – are not the correct starting point for finding the inverse. They don't represent the fundamental swap of x and y that's essential for this process. This highlights the importance of understanding the core concept behind finding inverses – it's not just about manipulating equations, it's about reversing the roles of input and output.

In essence, choosing the correct equation is about recognizing the first, most fundamental transformation needed to find the inverse. It's like choosing the right tool for the job – you need the one that gets the process started correctly. In this case, the swap of variables is that essential first step, making $x=5y^2+10$ the key to unlocking the inverse function.

Why the Other Options Don't Work

To really nail this concept, let's quickly address why the other options presented in the question don't work for finding the inverse. This will solidify your understanding and help you avoid similar mistakes in the future.

  • $\frac{1}{y}=5x^2+10$: This equation involves the reciprocal of y, but it doesn't swap x and y. It's changing the output of the original function, but not setting up the inverse relationship where we're trying to find the input that corresponds to a given output. This equation might be useful in a different context, but not for finding inverses.
  • $-y=5x^2+10$: This equation simply negates y. Again, it doesn't involve swapping x and y, so it doesn't represent the inverse relationship. It's a transformation of the original function, but not the one we need to reverse the process.
  • $y=\frac{1}{5}x^2+\frac{1}{10}$: This equation looks like it might be trying to