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The captivating allure of mathematics lies in exploring the inverse equation of **y = 9x² – 4**. By unraveling the **inverse** of a function, mathematicians can unlock a hidden world where the roles of input and output are **reversed**, unveiling new insights and possibilities.

Among the** myriad functions** that have captured the attention of **mathematicians**, the **inverse** of **y=9x² – 4** stands as a **captivating puzzle**.

In this article, we embark on a journey into the depths of this **inverse**, delving into the intricate processes of **reflection**, **transformation**, and mathematical **reversals**. Join us as we traverse the fascinating terrain of the **inverse** of **y=9x² – 4**, where mathematical mysteries await **unraveling**.

**Defining the Inverse Equation of ****y = 9x² – 4**

**y = 9x² – 4**

The **inverse** of a function is a **mathematical operation** that **undoes** the original function, effectively **swapping** the roles of the input and output variables. In the case of the **inverse** of **y = 9x² – 4**, we aim to find a new function that, when **applied** to the output values of the original function, yields the **corresponding input values**. In other words, we seek a function that, when applied to **y**, will give us the corresponding** x** values that satisfy the equation. Below, we present the graphical representation of the function **y = 9x² – 4** in Figure-1.

Figure-1.

** Mathematically**, the **inverse** of **y = 9x² – 4** is denoted as** x = (√(y+4))/3** or **x = – (√(y+4))/3**. The **inverse** function enables us to explore the **relationship** between the output and input variables from a different perspective. It provides a powerful tool for solving equations and **analyzing** the behavior of the original function.

## Finding the Inverse of **y = 9x² – 4**

To find the inverse of the function** y = 9x² – 4**, we follow these steps:

### Step 1

**Replace y** with **x** and **x** with **y**: **Swap** the variables** x** and **y** in the original equation, giving us the equation **x = 9y² – 4**.

### Step 2

Solve the **equation** for** y**: **Rearrange** the equation to **isolate y**. In this case, we have:

x = 9y² – 4

x + 4 = 9y²

(1/9)(x + 4) = y²

√((1/9)(x + 4)) = y

### Step 3

Consider the **positive** and **negative** **square root**: The equation above has two solutions, taking the positive and negative square root. Therefore, the **inverse function** has two branches: y₁ = √((1/9)(x + 4))

y₂ = -√((1/9)(x + 4))

### Step 4

Write the i**nverse function**: Combine the branches to express the inverse function in a **general form**. The inverse of **y = 9x² – 4** is given by:

f⁻¹(x) = √((1/9)(x + 4))

and:

f⁻¹(x) = -√((1/9)(x + 4))

The **inverse function** allows us to find the original input values **(x)** corresponding to given output values **(y).** By applying the inverse function to a given y, we can determine the corresponding **x** values that satisfy the **equation**. Below, we present the graphical representation of the inverse of the function **y = 9x² – 4** in Figure-2.

Figure-2.

## Applications

The **inverse** of the function **y = 9x² – 4** has various applications across different fields of **mathematics** and beyond. Here are some notable examples:

### Function Reversal and Solving Equations

The **inverse function** enables us to reverse the roles of **input** and **output** variables. In this case, the **inverse function** allows us to solve equations involving the **original function**. By finding the **inverse** of **y = 9x² – 4**, we can determine the **input values (x)** corresponding to specific **output values (y)**. This is particularly useful in solving equations where the **dependent variable** is given, and we need to find the corresponding **independent variable**.

### Curve Sketching and Transformation

The **inverse function** helps analyze the shape and behavior of the **original function**. By examining the graph of the **inverse function**, we can understand the **symmetry** and **transformation** properties of the **original function**** y = 9x² – 4**. In particular, the **inverse function** may reveal insights into the **original function’s** **concavity**, **intercepts**, **turning points**, and other characteristics.

### Optimization and Critical Points

In **optimization problems**, the **inverse function** can aid in identifying **critical points**. By analyzing the **inverse function**, we can determine the **input values (x)** that yield **extreme output values (y)**. This can be valuable in various applications, such as finding a quantity’s **maximum** or **minimum values**.

### Data Analysis and Modeling

The **inverse function** can be employed in **data analysis** and **modeling** to understand the relationship between variables. By finding the **inverse** of a **mathematical model**, we can obtain an explicit formula for the **dependent variable** as a function of the **independent variable**. This allows for better interpretation of the data and facilitates **predictions** or **estimations** based on the model.

### Physics and Engineering

The **inverse function** has practical applications in **physics** and **engineering**, where mathematical relationships are often encountered. For example, in **motion problems**, the **inverse function** can be used to determine the **time** needed to reach a specific position given the **displacement function**. In **electrical engineering**, the **inverse function** can help solve circuit **voltage**, **current**, and **resistance problems**.

### Computer Graphics and Animation

The **inverse function** finds application in **computer graphics** and **animation**, specifically in **transformations** and **deformations**. By using the **inverse function**, designers and animators can manipulate objects and characters to achieve desired effects, such as **scaling**, **rotation**, or **morphing**.

**Exercise **

### Example 1

Find the inverse function of** y = 9x² – 4** and determine its** domain** and **range**.

### Solution

To find the inverse function, we follow the steps mentioned earlier. First, we swap **x** and** y**:

**x = 9y² – 4**

Next, we solve for y:

x + 4 = 9y²

(1/9)(x + 4) = y

So, the inverse function is: f⁻¹(x) = (1/9)(x + 4)

The **domain** of the inverse function is the set of all **real numbers** since there are no restrictions on **x**. The **range** of the inverse function is also the set of all **real numbers**, as every real number can be obtained by substituting values into the **inverse function**.

### Example 2

Find the inverse function of** y = 3x² + 2**

### Solution

To find the inverse function of y = 3x² + 2, we can follow the steps outlined earlier:

Step 1: Swap **x** and **y**:

x = 3y² + 2

Step 2: Solve for **y**:

Rearrange the equation to **isolate** **y**. In this case, we have:

3y² = x – 2

y² = (x – 2) / 3

y = ±√((x – 2) / 3)

Step 3: Combine the branches: Since we have a **square root**, we must consider both the **positive** and **negative branches**. Therefore, the inverse function has two branches:

f⁻¹(x) = √((x – 2) / 3)

and:

f⁻¹(x) = -√((x – 2) / 3)

Figure-3.

### Example 3

Find the inverse function of** y = 2x² + 4x – 1**

### Solution

To find the inverse function of y = 2x² + 4x – 1, we can follow the same steps as before:

Step 1: Swap x and y:

x = 2y² + 4y – 1

Step 2: Solve for** y**: Rearrange the equation to isolate **y**. In this case, we have a quadratic equation:

2y² + 4y – 1 = x

To solve this **quadratic equation** for **y**, we can use the **quadratic formula**:

y = (-b ± √(b² – 4ac)) / (2a)

In this case, **a = 2**, **b = 4**, and **c = -1**. Substituting these values into the quadratic formula, we get:

y = (-4 ± √(4² – 4(2)(-1))) / (2(2))

y = (-4 ± √(16 + 8)) / 4

y = (-4 ± √24) / 4

y = (-4 ± 2√6) / 4

y = -1 ± (√6) / 2

So, the **inverse function** has two branches:

f⁻¹(x) = (-1 + √6) / 2

and:

f⁻¹(x) = (-1 – √6) / 2

Figure-4.

*All images were created with MATLAB.*