Find The Inverse Of The Function Y X2 12

Uncover the Mystery Behind Inverse Functions: Exploring y = x^2 – 12

In the realm of mathematics, where functions dance gracefully, there lies a task that often evokes a sense of trepidation among students: finding the inverse of a function. When confronted with the enigmatic equation y = x^2 – 12, many falter, their minds reeling with uncertainty. But fear not, for within these depths of mathematical intrigue lies a path to enlightenment.

The inverse of a function, like a mirror image or a doppelgänger, possesses a unique characteristic: when graphed, it creates a reflection of the original function across the diagonal line y = x. Determining the inverse of y = x^2 – 12, therefore, demands a methodical approach and a keen understanding of the underlying concepts.

Unveiling the Inverse of y = x^2 – 12

To embark on this mathematical adventure, let us begin by swapping the roles of x and y:

x = y^2 - 12

Next, we isolate y by performing the following algebraic gymnastics:

y^2 = x + 12
y = ±√(x + 12)

Thus, the inverse of y = x^2 – 12 is:

y = √(x + 12)

or

y = -√(x + 12)

Distilling the Essence of Inverse Functions

In essence, finding the inverse of a function involves two crucial steps: interchanging x and y and isolating one variable. By unraveling the mystery of y = x^2 – 12, we have not only solved a mathematical puzzle but also gained invaluable insights into the fascinating world of inverse functions. Remember, with patience and perseverance, any mathematical enigma can be conquered.

Find The Inverse Of The Function Y X2 12

Finding the Inverse of the Function y = x^2 – 12

Introduction

In mathematics, finding the inverse of a function involves switching the roles of the independent and dependent variables. This article aims to guide you through the steps to find the inverse of the function y = x^2 – 12.

Prerequisites

Before delving into the inverse function, a fundamental understanding of the following concepts is essential:

  • Functions and their notation
  • Graph interpretation
  • Algebraic operations

Steps to Find the Inverse

Step 1: Swap Variables

The first step is to interchange the variables x and y in the original function:

<center><img src="https://tse1.mm.bing.net/th?q=Step+1%3A+Swap+Variables" alt="Step 1: Swap Variables"></center>

x = y^2 – 12

Step 2: Solve for y

Next, we need to solve the equation for y. Isolate y by adding 12 to both sides and taking the square root:

<center><img src="https://tse1.mm.bing.net/th?q=Step+2%3A+Solve+for+y" alt="Step 2: Solve for y"></center>

y = ±√(x + 12)

Step 3: Define the Inverse

The inverse function is denoted as y^-1(x) or f^-1(x). By substituting y from the previous step into the original equation, we get:

<center><img src="https://tse1.mm.bing.net/th?q=Step+3%3A+Define+the+Inverse" alt="Step 3: Define the Inverse"></center>

y^-1(x) = ±√(x + 12)

Important Note:

The inverse function produces two solutions, one positive and one negative. This is because the original function is not one-to-one (fails the horizontal line test).

Domain and Range

The domain of the inverse function is the range of the original function, which is [0, ∞). The range of the inverse function is the domain of the original function, which is [-6, ∞).

Graph of the Inverse

The graph of the inverse function is a reflection of the graph of the original function across the line y = x.

<center><img src="https://tse1.mm.bing.net/th?q=Graph+of+the+Inverse" alt="Graph of the Inverse"></center>

Conclusion

By following the steps outlined above, we have successfully found the inverse of the function y = x^2 – 12. The inverse function is y^-1(x) = ±√(x + 12), and it reflects the original function across the line y = x.

FAQs

  1. What is the domain of the inverse function y^-1(x)?
  • [0, ∞)
  1. What is the range of the inverse function y^-1(x)?
  • [-6, ∞)
  1. Why does the inverse function have two solutions?
  • Because the original function is not one-to-one.
  1. Is the inverse function of a quadratic function also quadratic?
  • Yes, if the quadratic function is one-to-one.
  1. How can I determine if a function is one-to-one?
  • By using the horizontal line test or by verifying that the function is strictly increasing or decreasing.

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