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

by Alex Johnson 48 views

In mathematics, finding the inverse of a function is a common task, especially when dealing with quadratic equations. The inverse function essentially "undoes" what the original function does. In this article, we will explore how to find the inverse of the function y = 5x² + 10, breaking down the process step by step and discussing the correct equation to simplify. Let's dive in and make this concept crystal clear!

Understanding Inverse Functions

Before we jump into the specifics, let's quickly recap what an inverse function is. In simple terms, if a function f takes an input x and produces an output y, the inverse function, denoted as f⁻¹, takes y as input and produces x as output. Mathematically, if y = f(x), then x = f⁻¹(y). The key to finding an inverse function is to swap the roles of x and y and then solve for y.

The Process of Finding an Inverse

To find the inverse of a function, we generally follow these steps:

  1. Swap x and y: Replace every y with x and every x with y in the original equation.
  2. Solve for y: Rearrange the equation to isolate y on one side. This new equation represents the inverse function.
  3. Notation: Replace y with f⁻¹(x) to denote the inverse function.

This process might seem abstract, but it becomes clearer with examples. So, let's apply these steps to our given function, y = 5x² + 10.

Applying the Steps to y = 5x² + 10

Now, let's apply these steps to find the inverse of the function y = 5x² + 10. This will help us identify the correct equation that can be simplified to find the inverse.

  1. Swap x and y

The first step is to swap x and y in the equation. So, wherever we see y, we replace it with x, and wherever we see x, we replace it with y. This gives us:

x = 5y² + 10

This is the crucial step in finding the inverse because it sets up the equation that we need to solve for y.

  1. Solve for y

Now, we need to isolate y in the equation x = 5y² + 10. This involves a few algebraic steps. First, we subtract 10 from both sides:

x - 10 = 5y²

Next, we divide both sides by 5:

(x - 10) / 5 = y²

To get y by itself, we take the square root of both sides. Remember that taking the square root gives us both positive and negative solutions:

y = ±√((x - 10) / 5)

  1. Notation

Finally, we can write the inverse function using the proper notation. We replace y with f⁻¹(x):

f⁻¹(x) = ±√((x - 10) / 5)

This is the inverse function of y = 5x² + 10. The ± sign indicates that there are two possible values for the inverse, reflecting the symmetry of the original quadratic function.

Identifying the Correct Initial Equation

Looking back at the steps, the equation we arrived at immediately after swapping x and y was x = 5y² + 10. This is the equation that we then simplified to find the inverse. Therefore, the correct equation that can be simplified to find the inverse of y = 5x² + 10 is x = 5y² + 10.

Analyzing the Given Options

Now that we've walked through the process of finding the inverse, let's look at the given options and see which one matches our initial step.

  • A. x = 5y² + 10
  • B. 1/y = 5x² + 10
  • C. -y = 5x² + 10
  • D. y = (1/5)x² + (1/10)

Option A: x = 5y² + 10

As we determined earlier, the first step in finding the inverse is to swap x and y. Applying this to y = 5x² + 10 gives us x = 5y² + 10. This is exactly what Option A states. Therefore, Option A is the correct starting point for finding the inverse.

This option correctly represents the first step in finding the inverse function. By swapping x and y, we set up the equation that needs to be solved for y to find the inverse. This equation captures the essence of reversing the roles of input and output, which is fundamental to the concept of inverse functions. The subsequent steps of isolating y involve algebraic manipulations, but this initial swap is the foundation upon which the rest of the process rests.

Option B: 1/y = 5x² + 10

Option B, 1/y = 5x² + 10, is not the correct first step. This equation represents the reciprocal of y rather than swapping x and y. Taking the reciprocal of a function is a different operation than finding its inverse.

Option C: -y = 5x² + 10

Option C, -y = 5x² + 10, involves negating y but does not swap x and y. This equation represents a reflection of the original function across the x-axis, which is not the same as finding the inverse.

Option D: y = (1/5)x² + (1/10)

Option D, y = (1/5)x² + (1/10), looks like an attempt to manipulate the original equation, but it doesn't correctly represent the inverse. This equation alters the coefficients but doesn't perform the necessary swap of variables.

Detailed Explanation of the Correct Option

Option A, x = 5y² + 10, is the equation that can be simplified to find the inverse of y = 5x² + 10. Let's delve deeper into why this is the case.

The inverse function is found by reversing the roles of x and y. This means that the input of the original function becomes the output of the inverse function, and vice versa. Mathematically, this is achieved by swapping x and y in the original equation.

In the equation y = 5x² + 10, x is the independent variable (input), and y is the dependent variable (output). To find the inverse, we treat y as the input and x as the output. This is exactly what we do when we swap x and y.

Step-by-Step Simplification of x = 5y² + 10

To further illustrate this, let's go through the steps of simplifying x = 5y² + 10 to solve for y:

  1. Subtract 10 from both sides: This isolates the term with y².

    x - 10 = 5y²

  2. Divide both sides by 5: This isolates y².

    (x - 10) / 5 = y²

  3. Take the square root of both sides: This solves for y. Remember to include both positive and negative roots.

    y = ±√((x - 10) / 5)

This final equation, y = ±√((x - 10) / 5), represents the inverse function. By starting with x = 5y² + 10, we have successfully set up the equation that, when solved for y, gives us the inverse.

Conclusion

In conclusion, the equation that can be simplified to find the inverse of y = 5x² + 10 is A. x = 5y² + 10. This is because the fundamental step in finding the inverse is to swap x and y, which is exactly what this equation represents. By understanding the process of swapping variables and then solving for y, we can confidently find the inverse of various functions.

Understanding inverse functions is a critical skill in mathematics, and mastering the steps to find them will help in more advanced topics. Remember, the key is to swap x and y and then isolate y. Keep practicing, and you'll become proficient in finding inverses!

For further reading and to deepen your understanding of inverse functions, you might find helpful resources at Khan Academy's Inverse Functions Section.