## 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.

## Navigating the Perplexities of Inverse Functions

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.

**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**

**What is the domain of the inverse function y^-1(x)?**

- [0, ∞)

**What is the range of the inverse function y^-1(x)?**

- [-6, ∞)

**Why does the inverse function have two solutions?**

- Because the original function is not one-to-one.

**Is the inverse function of a quadratic function also quadratic?**

- Yes, if the quadratic function is one-to-one.

**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.

.

Find,Inverse,Function