Finding The Inverse Of $y = 9x^2 - 4$ A Step-by-Step Guide

Understanding Inverse Functions

In mathematics, an inverse function is a function that "reverses" the effect of another function. If a function f takes an input x and produces an output y, then the inverse function, denoted as f⁻¹*, takes the output y and returns the original input x. In simpler terms, the inverse function "undoes" what the original function does. Finding the inverse of a function involves a systematic process of swapping the roles of the independent variable (x) and the dependent variable (y) and then solving for y. This process effectively reverses the mapping performed by the original function. However, it is crucial to note that not all functions have inverses. For a function to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto). Injectivity ensures that each output corresponds to a unique input, while surjectivity ensures that every element in the codomain has a corresponding element in the domain. In the context of quadratic functions, like the one we're addressing, the existence of an inverse is conditional. Quadratic functions, due to their parabolic nature, typically do not pass the horizontal line test, which is a visual method to check for injectivity. To find an inverse, we often need to restrict the domain of the original quadratic function to an interval where it is either strictly increasing or strictly decreasing, ensuring it becomes one-to-one within that restricted domain. This restriction allows us to define a valid inverse function that maps the range of the restricted domain back to the domain.

The Given Equation: $y = 9x^2 - 4$

Our main goal is to determine which equation represents the inverse of the quadratic equation $y = 9x^2 - 4$. This equation defines a parabola that opens upwards. The key to finding the inverse lies in swapping the variables x and y and then isolating y. This is because the inverse function essentially reverses the roles of input and output. The given equation is a standard quadratic form, and its graph is a parabola with its vertex at (0, -4). The parabola opens upwards because the coefficient of the $x^2$ term is positive (9 > 0). This is crucial because the parabolic nature of the function implies that it is not one-to-one over its entire domain, which means it does not have a simple inverse. However, we can find an inverse if we restrict the domain to either $x \geq 0$ or $x \leq 0$. This is because, within these restricted domains, the function becomes one-to-one, and an inverse can be defined. Without restricting the domain, the inverse relation will not be a function, but rather a relation that maps one input to multiple outputs. In solving for the inverse, we will encounter a square root, which inherently introduces a $\pm$ sign, reflecting the two possible branches of the inverse relation corresponding to the two halves of the parabola. The constant term -4 in the original equation will play a role in shifting the graph of the inverse function, and the coefficient 9 will affect the scaling. By carefully manipulating the equation after swapping x and y, we will arrive at the correct expression for the inverse, keeping in mind the necessity of the $\pm$ sign due to the nature of the square root operation. This process will allow us to identify which of the provided options correctly represents the inverse of the given quadratic equation.

Step-by-Step Solution: Finding the Inverse

To find the inverse of the equation $y = 9x^2 - 4$, we'll follow these steps:

1. Swap x and y: This is the foundational step in finding the inverse. It reflects the reversal of the input and output roles. Replacing y with x and x with y gives us:
   $x = 9y^2 - 4$

2. Isolate the $y^2$ term: We aim to get the term containing $y^2$ by itself on one side of the equation. Add 4 to both sides: $x + 4 = 9y^2$

3. Divide by the coefficient of $y^2$: The coefficient of $y^2$ is 9. Divide both sides by 9 to further isolate $y^2$: $\frac{x + 4}{9} = y^2$

4. Take the square root of both sides: To solve for y, we need to eliminate the square. Taking the square root introduces both positive and negative solutions, which is crucial for the inverse of a quadratic function: $y = \pm \sqrt{\frac{x + 4}{9}}$

5. Simplify the square root: We can simplify the square root by recognizing that the square root of a quotient is the quotient of the square roots, and the square root of 9 is 3: $y = \frac{\pm \sqrt{x + 4}}{\sqrt{9}}$ $y = \frac{\pm \sqrt{x + 4}}{3}$

This final equation represents the inverse of the original equation. The $\pm$ sign indicates that for each x value, there are two corresponding y values (except when $x = -4$), which is typical for the inverse of a quadratic function due to its parabolic nature. This step-by-step process ensures that we accurately reverse the operations performed by the original function, leading us to the correct inverse function. Each step is carefully chosen to isolate y while adhering to the rules of algebra, ultimately revealing the relationship that "undoes" the original quadratic equation.

Analyzing the Answer Choices

Now, let's compare our solution, $y = \frac{\pm \sqrt{x + 4}}{3}$, with the provided options:

A. $y = \frac{\pm \sqrt{x + 4}}{9}$ B. $y = \pm \sqrt{\frac{x}{9} + 4}$ C. $y = \frac{\pm \sqrt{x + 4}}{3}$ D. $y = \frac{\pm \sqrt{x}}{3} + \frac{2}{3}$

Upon comparing our derived inverse function with the given options, it's evident that option C, $y = \frac{\pm \sqrt{x + 4}}{3}$, exactly matches our solution. This confirms that the correct inverse function is indeed represented by option C. Options A, B, and D, on the other hand, differ in their forms. Option A has a denominator of 9 instead of 3, indicating an incorrect simplification of the square root. Option B has the terms inside the square root incorrectly manipulated, suggesting a misunderstanding of how the inverse transformation affects the equation's structure. Option D presents a completely different form, which does not align with the step-by-step algebraic manipulation we performed to find the inverse. The careful comparison process ensures that we select the option that precisely represents the inverse relationship we derived, highlighting the importance of accurately executing each step in the solution and recognizing the correct algebraic form of the inverse function.

Therefore, the correct answer is C. y=\frac{ rac{+}{-} rac{}{x+4}}{3}

This comprehensive solution demonstrates how to find the inverse of a quadratic equation, emphasizing the crucial steps of variable swapping, algebraic manipulation, and simplification. It highlights the importance of understanding the properties of inverse functions and the significance of the $\pm$ sign when dealing with square roots. By following this methodical approach, one can confidently tackle similar problems involving inverse functions.