Which Equation Can Be Simplified To Find The Inverse Of $y=2x^2$?A. $\frac{1}{y}=2x^2$ B. $y=\frac{1}{2}x^2$ C. $-y=2x^2$ D. $x=2y^2$

Introduction

In mathematics, finding the inverse of a function is a crucial concept that helps us understand the relationship between the input and output values of a function. The inverse of a function essentially reverses the operation of the original function. In this article, we will explore how to find the inverse of a quadratic equation, specifically the equation $y=2x^2$. We will examine the given options and determine which one can be simplified to find the inverse of the given equation.

Understanding the Concept of Inverse Functions

Before we dive into finding the inverse of the given equation, let's briefly review the concept of inverse functions. An inverse function is a function that reverses the operation of the original function. In other words, if we have a function $f(x)$, its inverse function is denoted as $f^{-1}(x)$ and satisfies the property:

$f(f^{-1}(x)) = x$

In other words, applying the inverse function to the output of the original function returns the original input.

Finding the Inverse of a Quadratic Equation

To find the inverse of a quadratic equation, we need to follow a specific procedure. The general form of a quadratic equation is:

$y = ax^2 + bx + c$

To find the inverse of this equation, we need to swap the variables $x$ and $y$ and then solve for $y$. This will give us the inverse function.

Step 1: Swap the Variables

The first step in finding the inverse of a quadratic equation is to swap the variables $x$ and $y$. This means that we replace $x$ with $y$ and $y$ with $x$ in the original equation.

Step 2: Solve for y

Once we have swapped the variables, we need to solve for $y$. This involves isolating $y$ on one side of the equation.

Step 3: Simplify the Equation

After solving for $y$, we need to simplify the equation to obtain the inverse function.

Applying the Procedure to the Given Equation

Now that we have reviewed the procedure for finding the inverse of a quadratic equation, let's apply it to the given equation $y=2x^2$.

Step 1: Swap the Variables

Swapping the variables $x$ and $y$ in the given equation, we get:

$x = 2y^2$

Step 2: Solve for y

To solve for $y$, we need to isolate $y$ on one side of the equation. We can do this by taking the square root of both sides of the equation:

$y = \pm \sqrt{\frac{x}{2}}$

Step 3: Simplify the Equation

The equation $y = \pm \sqrt{\frac{x}{2}}$ is the inverse of the given equation $y=2x^2$. However, we need to determine which of the given options can be simplified to obtain this equation.

Evaluating the Options

Let's examine each of the given options and determine which one can be simplified to obtain the inverse of the given equation.

Option A: $\frac{1}{y}=2x^2$

To evaluate this option, we can start by taking the reciprocal of both sides of the equation:

$\frac{1}{y} = 2x^2$

This equation is not in the form of the inverse of the given equation $y=2x^2$.

Option B: $y=\frac{1}{2}x^2$

This option is not in the form of the inverse of the given equation $y=2x^2$.

Option C: $-y=2x^2$

To evaluate this option, we can start by multiplying both sides of the equation by $-1$:

$y = -2x^2$

This equation is not in the form of the inverse of the given equation $y=2x^2$.

Option D: $x=2y^2$

This option is in the form of the inverse of the given equation $y=2x^2$. By swapping the variables $x$ and $y$, we can obtain the inverse function:

$y = \pm \sqrt{\frac{x}{2}}$

This is the same equation we obtained in Step 3 of the procedure for finding the inverse of a quadratic equation.

Conclusion

In conclusion, the correct option is D. $x=2y^2$. This option can be simplified to obtain the inverse of the given equation $y=2x^2$. By following the procedure for finding the inverse of a quadratic equation, we can determine that the inverse of the given equation is $y = \pm \sqrt{\frac{x}{2}}$.