Which Equation Is The Inverse Of Y = 9 X 2 − 4 Y=9x^2-4 Y = 9 X 2 − 4 ?A. Y = ± X + 4 9 Y=\frac{\pm \sqrt{x+4}}{9} Y = 9 ± X + 4 ​ ​ B. Y = ± X 9 + 4 Y=\pm \sqrt{\frac{x}{9}+4} Y = ± 9 X ​ + 4 ​ C. Y = ± X + 4 3 Y=\frac{\pm \sqrt{x+4}}{3} Y = 3 ± X + 4 ​ ​ D. Y = ± X 3 + 2 3 Y=\frac{\pm \sqrt{x}}{3}+\frac{2}{3} Y = 3 ± X ​ ​ + 3 2 ​

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Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. In this article, we will focus on finding the inverse of a quadratic equation, specifically the equation y = 9x^2 - 4.

What is an Inverse Equation?

An inverse equation is a function that undoes the operation of another function. In the case of a quadratic equation, the inverse equation will have the form y = f^(-1)(x), where f(x) is the original quadratic equation. To find the inverse of a quadratic equation, we need to swap the variables x and y and then solve for y.

Finding the Inverse of a Quadratic Equation

To find the inverse of a quadratic equation, we can follow these steps:

  1. Swap the variables x and y: This means that we will replace x with y and y with x in the original equation.
  2. Solve for y: Once we have swapped the variables, we need to solve for y in terms of x.

Step 1: Swap the Variables x and y

Let's start by swapping the variables x and y in the original equation y = 9x^2 - 4. This gives us:

x = 9y^2 - 4

Step 2: Solve for y

Now that we have swapped the variables, we need to solve for y in terms of x. To do this, we can add 4 to both sides of the equation and then divide by 9:

x + 4 = 9y^2

y^2 = (x + 4) / 9

y = ±√((x + 4) / 9)

Simplifying the Inverse Equation

We can simplify the inverse equation by multiplying both sides by √9:

y = ±√(x + 4) / 3

This is the inverse equation of the original quadratic equation y = 9x^2 - 4.

Comparing the Options

Now that we have found the inverse equation, let's compare it with the options given:

A. y = ±√(x + 4) / 9 B. y = ±√((x + 4) / 9) C. y = ±√(x + 4) / 3 D. y = ±√(x) / 3 + 2/3

The correct answer is C. y = ±√(x + 4) / 3.

Conclusion

In this article, we have discussed the concept of inverse equations and how to find the inverse of a quadratic equation. We have also compared the options given and found that the correct answer is C. y = ±√(x + 4) / 3. We hope that this article has provided a comprehensive guide to inverse equations and has helped you to understand the concept better.

Frequently Asked Questions

  • What is an inverse equation? An inverse equation is a function that reverses the operation of another function.
  • How do I find the inverse of a quadratic equation? To find the inverse of a quadratic equation, you need to swap the variables x and y and then solve for y.
  • What is the inverse equation of y = 9x^2 - 4? The inverse equation of y = 9x^2 - 4 is y = ±√(x + 4) / 3.

References

Glossary

  • Inverse function: A function that reverses the operation of another function.
  • Quadratic equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Inverse equation: A function that undoes the operation of another function.
    Inverse Equations: A Comprehensive Guide =====================================================

Q&A: Inverse Equations

Q: What is an inverse equation? A: An inverse equation is a function that reverses the operation of another function. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Q: How do I find the inverse of a quadratic equation? A: To find the inverse of a quadratic equation, you need to swap the variables x and y and then solve for y. This involves adding 4 to both sides of the equation, dividing by 9, and then taking the square root of both sides.

Q: What is the inverse equation of y = 9x^2 - 4? A: The inverse equation of y = 9x^2 - 4 is y = ±√(x + 4) / 3.

Q: Can I use a calculator to find the inverse of a quadratic equation? A: Yes, you can use a calculator to find the inverse of a quadratic equation. However, it's always a good idea to check your work by plugging the inverse equation back into the original equation to make sure it's correct.

Q: What is the difference between an inverse function and an inverse equation? A: An inverse function is a function that reverses the operation of another function, while an inverse equation is a specific type of inverse function that is used to solve quadratic equations.

Q: Can I use the inverse of a quadratic equation to solve a system of equations? A: Yes, you can use the inverse of a quadratic equation to solve a system of equations. However, you need to make sure that the inverse equation is correct and that it's used correctly in the system of equations.

Q: What are some common mistakes to avoid when finding the inverse of a quadratic equation? A: Some common mistakes to avoid when finding the inverse of a quadratic equation include:

  • Swapping the variables x and y incorrectly
  • Not adding 4 to both sides of the equation
  • Not dividing by 9
  • Not taking the square root of both sides
  • Not checking the work by plugging the inverse equation back into the original equation

Q: Can I use the inverse of a quadratic equation to solve a rational equation? A: Yes, you can use the inverse of a quadratic equation to solve a rational equation. However, you need to make sure that the inverse equation is correct and that it's used correctly in the rational equation.

Q: What is the inverse of a rational equation? A: The inverse of a rational equation is a function that reverses the operation of the rational equation. It's used to solve rational equations and is an important concept in algebra.

Q: Can I use the inverse of a quadratic equation to solve a polynomial equation? A: Yes, you can use the inverse of a quadratic equation to solve a polynomial equation. However, you need to make sure that the inverse equation is correct and that it's used correctly in the polynomial equation.

Q: What is the inverse of a polynomial equation? A: The inverse of a polynomial equation is a function that reverses the operation of the polynomial equation. It's used to solve polynomial equations and is an important concept in algebra.

Conclusion

In this article, we have discussed the concept of inverse equations and how to find the inverse of a quadratic equation. We have also answered some common questions about inverse equations and provided some tips and tricks for finding the inverse of a quadratic equation. We hope that this article has provided a comprehensive guide to inverse equations and has helped you to understand the concept better.

Frequently Asked Questions

  • What is an inverse equation? An inverse equation is a function that reverses the operation of another function.
  • How do I find the inverse of a quadratic equation? To find the inverse of a quadratic equation, you need to swap the variables x and y and then solve for y.
  • What is the inverse equation of y = 9x^2 - 4? The inverse equation of y = 9x^2 - 4 is y = ±√(x + 4) / 3.

References

Glossary

  • Inverse function: A function that reverses the operation of another function.
  • Quadratic equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Inverse equation: A function that undoes the operation of another function.
  • Rational equation: An equation that contains a fraction with polynomials in the numerator and denominator.
  • Polynomial equation: An equation that contains a polynomial expression.