Inverse Of Y=9x^2-4: A Step-by-Step Solution

Hey guys! Today, we're diving into a super important concept in math: finding the inverse of a function. Specifically, we're going to tackle the equation $y = 9x^2 - 4$ and figure out which of the given options represents its inverse. This is a fundamental skill in algebra and calculus, so let’s break it down step by step to make sure we’ve got it nailed. So, let’s dive in and make math a little less mysterious, shall we?

Understanding Inverse Functions

Before we jump into the nitty-gritty, let's quickly recap what an inverse function actually is. Think of a function like a machine: you put something in (an input, usually 'x'), and it spits something else out (an output, usually 'y'). The inverse function is like reversing that machine. You put the output back in, and it gives you the original input. Mathematically, if $f(x)$ gives you $y$, then the inverse function, often written as $f^{-1}(x)$, takes $y$ and gives you back $x$.

To find the inverse, there's a simple two-step process we usually follow:

1. Swap x and y: Replace every 'y' with an 'x' and every 'x' with a 'y' in your equation.
2. Solve for y: Rearrange the equation to isolate 'y' on one side. This new 'y' is your inverse function!

Why is this important? Well, inverse functions help us undo operations. They're used in cryptography, solving equations, and understanding relationships between different mathematical models. Plus, knowing how to find them is a key building block for more advanced math topics. For example, in calculus, understanding inverse functions is crucial for working with inverse trigonometric functions and exponential and logarithmic functions. So, mastering this now will definitely pay off later!

Step-by-Step Solution for $y = 9x^2 - 4$

Alright, let's get our hands dirty and find the inverse of $y = 9x^2 - 4$. Remember those steps we just talked about? Let's put them into action.

1. Swap x and y

Our original equation is $y = 9x^2 - 4$. The first step is to swap 'x' and 'y'. This gives us:

$x = 9y^2 - 4$

Yep, it's that simple! We've just interchanged the roles of input and output. This single step is the heart of finding an inverse function. It sets the stage for us to reverse the operations and isolate 'y'. It might seem like a small change, but it's a crucial one. By swapping 'x' and 'y', we're essentially looking at the equation from the opposite perspective, which is exactly what finding an inverse is all about.

2. Solve for y

Now comes the fun part: isolating 'y'. We need to get 'y' all by itself on one side of the equation. Let's do it step by step.

First, we want to get rid of that -4. We can do this by adding 4 to both sides of the equation:

$x + 4 = 9y^2$

Next, we need to deal with the 9 that's multiplying $y^2$. To undo multiplication, we divide. So, let's divide both sides by 9:

rac{x + 4}{9} = y^2

We're almost there! Now we have $y^2$, but we want 'y'. What's the opposite of squaring? Taking the square root! So, we take the square root of both sides:

y = rac{\pm \sqrt{x + 4}}{3}

Don't forget the ±! When we take the square root, we need to consider both the positive and negative solutions. This is super important because it reflects the fact that both a positive and a negative number, when squared, can give you the same positive result. For instance, both 3 squared and -3 squared equal 9. So, when we reverse the process, we need to account for both possibilities.

Analyzing the Options

Okay, we've found the inverse function: y = rac{\pm \sqrt{x + 4}}{3}. Now, let's compare this to the options given in the question.

We were given four 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. $y=\frac{ \pm \sqrt{x}}{3}+\frac{2}{3}$

By carefully comparing our solution with these options, we can see that option C, $y=\frac{ \pm \sqrt{x+4}}{3}$, perfectly matches what we found. Options A, B, and D have different arrangements of the terms and operations, making them incorrect.

Why Option C is the Correct Inverse

Option C, y = rac{\pm \sqrt{x + 4}}{3}, is the correct inverse because it is the result of correctly swapping $x$ and $y$ in the original equation $y = 9x^2 - 4$ and then accurately solving for $y$. Let’s recap the steps:

1. We started with the original equation: $y = 9x^2 - 4$.
2. We swapped $x$ and $y$ to get: $x = 9y^2 - 4$.
3. We added 4 to both sides: $x + 4 = 9y^2$.
4. We divided both sides by 9: $\frac{x + 4}{9} = y^2$.
5. We took the square root of both sides, remembering the $\pm$: $y = \frac{\pm \sqrt{x + 4}}{3}$.

This step-by-step process ensures that we have correctly reversed the operations performed in the original function. Each operation is undone in the reverse order, which is the essence of finding an inverse function. The $\pm$ sign is particularly important because it acknowledges that the original function, $y = 9x^2 - 4$, when reversed, can result in both positive and negative values for $y$. This is a characteristic feature of inverse functions involving squares or other even powers.

Common Mistakes to Avoid

Finding inverse functions can be a bit tricky, and there are some common pitfalls to watch out for. Let’s go over a few so you can steer clear of them!

Forgetting the ± When Taking Square Roots

As we saw in our solution, when you take the square root of both sides of an equation, you must remember to include both the positive and negative roots. For example, if you have $y^2 = 9$, then $y$ could be either 3 or -3. Forgetting the $\pm$ can lead to an incomplete or incorrect inverse function, especially when dealing with functions that involve even powers. This is because squaring both a positive and a negative number results in a positive number, so when we reverse the process, we need to consider both possibilities.

Incorrectly Applying Order of Operations

When solving for $y$ after swapping $x$ and $y$, it’s crucial to follow the correct order of operations (PEMDAS/BODMAS). Make sure you undo operations in the reverse order they were applied. For instance, in our example, we first added 4 to both sides before dividing by 9 because the subtraction of 4 was the last operation performed on $y$ in the original equation. Reversing this order is key to isolating $y$ correctly and finding the true inverse.

Not Swapping x and y at All

This might sound obvious, but it's a common mistake, especially when you're rushing through a problem. Remember, the very first step in finding an inverse is to swap $x$ and $y$. If you skip this step, you're not finding the inverse; you're just rearranging the original equation. Swapping $x$ and $y$ is what sets the stage for reversing the function's operations and finding its inverse.

Mixing Up the Steps

It’s easy to get the steps mixed up if you're not careful. Make sure you swap $x$ and $y$ first, and then solve for $y$. Doing it the other way around won't give you the correct inverse. This order is crucial because it reflects the fundamental idea of an inverse function: reversing the roles of input and output. Solving for $y$ before swapping would essentially be rearranging the original function, not finding its inverse.

Practice Makes Perfect

The best way to avoid these mistakes is to practice! The more you work through these types of problems, the more comfortable you'll become with the process. So, grab some more equations and try finding their inverses. You'll get the hang of it in no time!

Conclusion

So, to wrap things up, the correct answer to the question “Which equation is the inverse of $y = 9x^2 - 4$?” is C. y = rac{\pm \sqrt{x + 4}}{3}. We found this by swapping $x$ and $y$ and then carefully solving for $y$, remembering to include the $\pm$ when we took the square root. Finding inverse functions is a fundamental skill in mathematics, and understanding the steps and common pitfalls will help you succeed in algebra, calculus, and beyond. Keep practicing, and you'll master this concept in no time!

I hope this breakdown was helpful, guys! Keep up the awesome work, and happy math-ing!