To find the equation that represents the inverse of the given function y = 9x^2 – 4, you need to follow a series of steps. The inverse of a function essentially swaps the roles of x and y, so we’ll start by switching the variables:

- Replace y with x and x with y in the original equation: x = 9y^2 – 4
- Solve this equation for y to isolate the new y (which represents the inverse function): x = 9y^2 – 4

Add 4 to both sides of the equation: x + 4 = 9y^2

- Divide both sides by 9 to solve for y^2: (x + 4)/9 = y^2
- Take the square root of both sides to solve for y: y = ±√((x + 4)/9)

Now, you have the equation for the inverse function of y = 9x^2 – 4:

y = ±√((x + 4)/9)

The ± symbol indicates that there are two branches to the inverse function. One branch corresponds to the positive square root, and the other branch corresponds to the negative square root. Depending on the context and the specific application, you may use either branch or both in your calculations.

**Understanding the Inverse Function**

An inverse function undoes the action of the original function. In the context of y = 9x^2 – 4, this means that the inverse function will reverse the process of squaring and subtracting 4. To better grasp the concept, let’s break down the original and inverse functions.

**Original Function (y = 9x^2 – 4):**This function takes an input value x, squares it (multiplies by itself), multiplies the result by 9, and then subtracts 4. This is a parabolic function that opens upward, meaning it has a minimum point (the vertex) and extends infinitely in both directions along the x-axis.**Inverse Function (y = ±√((x + 4)/9)):**The inverse function undoes the operations of the original function. It starts by taking an input value x, then adds 4, divides the result by 9, and finally takes the square root. The ± symbol indicates that there are two possible solutions for each input x, corresponding to the positive and negative square roots. This implies that the inverse function is not a true function (in the strict mathematical sense) but rather a relation.

**Key Characteristics of the Inverse Function**

**Domain and Range:**The original function, y = 9x^2 – 4, has a domain of all real numbers (since you can square any real number) and a range of y ≥ -4 (the minimum value of the parabola). The inverse function, on the other hand, has a domain of x ≥ -4/9 (due to the square root of a non-negative number) and a range of all real numbers.**Symmetry:**The original function is symmetric with respect to the y-axis, meaning it has a central axis of symmetry. The inverse function retains this symmetry but is reflected across the line y = x, which is known as the identity line. This means that the inverse function is symmetric with respect to the identity line.**Inverse Operations:**The original function performs squaring (x^2) and then subtracting 4 (-4). In contrast, the inverse function reverses these operations by taking the square root (√) and adding 4. This is a fundamental property of inverse functions—they “undo” what the original function does.**Non-Unique Solution:**As mentioned earlier, the inverse function produces two possible values for each input x, one corresponding to the positive square root and the other to the negative square root. This non-uniqueness is a characteristic of square roots, as they can have positive and negative solutions.

**Practical Applications**

In real-world applications, the concept of inverse functions is commonly used in fields such as physics, engineering, finance, and computer science. In mathematics, inverse functions are essential for solving equations and finding solutions to problems that involve reversing mathematical operations.

For example, in physics, the inverse function might be used to determine the initial velocity of an object in free fall based on its final velocity and the time it took to fall. In finance, it could be used to calculate the original investment amount given the final balance and the interest rate. In computer science, inverse functions play a role in cryptography and data compression algorithms.

**Conclusion**

Understanding inverse functions is a fundamental concept in mathematics, and it has practical applications across various fields. In the case of the inverse of y = 9x^2 – 4, we’ve explored how it undoes the operations of the original function and the key characteristics that distinguish it. The non-uniqueness of solutions in the inverse function highlights the importance of considering both positive and negative branches when using it in practical applications.