Evaluate (f-g)(√-9) For F(x)=x^2+x & G(x)=3x+11

Hey guys! Today, we're diving into a fun math problem that involves evaluating a composite function with complex numbers. Specifically, we're going to figure out how to find the value of (f-g)(√-9), given that f(x) = x² + x and g(x) = 3x + 11. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step so it’s super easy to understand. Whether you're prepping for an exam or just love challenging your brain, this guide will walk you through the process. We'll cover everything from understanding function subtraction to dealing with imaginary numbers, ensuring you've got a solid grasp on the concepts. Let's get started and unravel this mathematical puzzle together! Remember, math can be fun, and with the right approach, even the trickiest problems become manageable. We will make sure every step is crystal clear, and by the end, you will be able to tackle similar problems with confidence. So, grab your pencils and notebooks, and let’s jump right in! Our goal is to not only solve this particular problem but also to equip you with the tools and understanding to handle any function evaluation that comes your way.

Understanding the Functions: f(x) and g(x)

Before we can tackle (f-g)(√-9), let's make sure we're crystal clear on what our functions, f(x) and g(x), actually mean. Understanding these functions is the bedrock of solving this problem, guys. So, let’s get this straight first. We have two functions here:

• f(x) = x² + x
• g(x) = 3x + 11

What do these function definitions really tell us? Well, a function is essentially a rule that takes an input (in this case, x) and spits out a corresponding output. Think of it like a machine: you feed it something, and it processes it according to a specific set of instructions. For f(x), the rule says, "Take the input x, square it, then add x to the result." So, if we were to plug in, say, x = 2 into f(x), we'd get f(2) = 2² + 2 = 4 + 2 = 6. See how that works? It’s all about following the rule. Similarly, g(x) has its own rule: "Take the input x, multiply it by 3, then add 11." If we plugged in x = 2 into g(x), we'd get g(2) = 3(2) + 11 = 6 + 11 = 17. So, each function has its own way of transforming the input x into an output. The key is to understand what these transformations are. Breaking down these functions like this makes it easier to see how they behave and how they will interact when we start combining them, like in the f-g operation we’re about to tackle. Understanding these basic rules is crucial because it sets the stage for more complex operations. Without a solid grasp of what f(x) and g(x) do individually, the rest of the problem will be much harder to navigate. We’re building a foundation here, so let’s make it strong! Now that we're clear on the individual functions, let's move on to the next step: understanding what it means to subtract one function from another. This is where things start to get really interesting, so stay with me! We are going to make sure that this concept clicks, because it’s a building block for even more advanced stuff later on.

Function Subtraction: Defining (f-g)(x)

Now that we understand our individual functions f(x) and g(x), let's tackle the concept of function subtraction, specifically what (f-g)(x) means. Function subtraction might sound fancy, but it's actually quite straightforward, guys. It’s like subtracting apples from oranges – well, kind of! In mathematical terms, (f-g)(x) simply means we're taking the entire function g(x) and subtracting it from the entire function f(x). Think of it as a pointwise operation: for any given x, we find the value of f(x), find the value of g(x), and then subtract the latter from the former. Mathematically, we can write it as: (f-g)(x) = f(x) - g(x). This is the core definition we need to keep in mind. So, what does this look like with our specific functions? We know that f(x) = x² + x and g(x) = 3x + 11. Plugging these into our definition, we get: (f-g)(x) = (x² + x) - (3x + 11). Now, we need to simplify this expression. Remember your algebra rules! We distribute the negative sign across the parentheses in g(x): (f-g)(x) = x² + x - 3x - 11. Next, we combine like terms: (f-g)(x) = x² + (x - 3x) - 11. This simplifies further to: (f-g)(x) = x² - 2x - 11. Voila! We've now found the new function (f-g)(x). This new function represents the result of subtracting g(x) from f(x) for any value of x. Understanding this step is crucial because it transforms the problem into a simpler evaluation. Instead of dealing with two separate functions, we now have a single function, (f-g)(x), that we can work with. This is a common technique in math: reducing complexity by combining terms and simplifying expressions. By performing this subtraction, we've essentially created a new tool that will help us solve the original problem. Remember, the goal here is not just to get the answer but to understand the process. So, let’s take a moment to appreciate what we've done. We've taken two functions, subtracted one from the other, and created a brand-new function that represents their difference. This is a powerful concept that you’ll use again and again in mathematics. Now that we have (f-g)(x), we’re one step closer to finding (f-g)(√-9). The next step is to understand how to deal with that square root of a negative number, which brings us into the fascinating world of imaginary numbers. So, let’s dive in!

Dealing with √-9: Introduction to Imaginary Numbers

Alright, guys, we've got (f-g)(x) = x² - 2x - 11, but our mission is to find (f-g)(√-9). That little √-9 is where things get interesting because it introduces us to the realm of imaginary numbers. Don't let the name scare you; they're not as mysterious as they sound! The key thing to remember is that the square root of a negative number is not a real number. Why? Because no real number, when multiplied by itself, gives a negative result. That's where imaginary numbers come in. The basic unit of imaginary numbers is i, which is defined as the square root of -1. So, i = √-1. This is our magic key to unlocking √-9. How do we use it? Well, we can rewrite √-9 as √(9 * -1). Using the properties of square roots, we can separate this into √9 * √-1. We know that √9 is 3, and we know that √-1 is i. Therefore, √-9 = 3i. Boom! We've just converted a square root of a negative number into an imaginary number. This is a crucial step because it allows us to work with these numbers in our calculations. Now, we know that the square root of -9 is 3i. This is a purely imaginary number because it's a multiple of i. It doesn't have a real part; it's just the imaginary part. This is an important distinction to make because when we eventually express our final answer in the form a + bi, we need to understand the real and imaginary components. Understanding imaginary numbers opens up a whole new world of mathematical possibilities. They're used extensively in various fields, from electrical engineering to quantum physics. So, mastering this concept is not just about solving this particular problem; it's about expanding your mathematical toolkit. Now that we've tamed the square root of -9 and transformed it into the manageable 3i, we're ready to plug it into our (f-g)(x) function. This is where the real fun begins – when we see how imaginary numbers interact with our algebraic expressions. So, are you ready to take the next step? Let's go!

Evaluating (f-g)(3i): Plugging in the Imaginary Value

Okay, guys, we've reached a pivotal point! We've determined that (f-g)(x) = x² - 2x - 11 and √-9 = 3i. Now, it's time to put it all together and evaluate (f-g)(3i). This is where the rubber meets the road, so let's take it step by step. Remember, evaluating a function at a particular value simply means substituting that value for x in the function's expression. In our case, we're substituting 3i for x in (f-g)(x) = x² - 2x - 11. So, we have: (f-g)(3i) = (3i)² - 2(3i) - 11. Now, let's break this down. First, we need to square 3i. Remember that (3i)² means (3i) * (3i). Multiplying this out, we get 9i². But what is i²? Recall that i = √-1. So, i² = (√-1)² = -1. This is a super important identity to remember when working with imaginary numbers! So, 9i² = 9(-1) = -9. Now we can substitute this back into our expression: (f-g)(3i) = -9 - 2(3i) - 11. Next, let's simplify the middle term: 2(3i) = 6i. So, our expression becomes: (f-g)(3i) = -9 - 6i - 11. Finally, we combine the real parts (-9 and -11): (f-g)(3i) = -20 - 6i. And there you have it! We've successfully evaluated (f-g)(3i). The result is -20 - 6i, which is a complex number in the form a + bi, where a is the real part (-20) and b is the imaginary part (-6). This step is crucial because it demonstrates how to handle imaginary numbers within algebraic expressions. It highlights the importance of knowing the basic properties of i, especially that i² = -1. By carefully substituting and simplifying, we've navigated through the imaginary numbers and arrived at a clear, concise answer. The process of plugging in 3i might have seemed a bit daunting at first, but by breaking it down into smaller steps, we've made it manageable and even, dare I say, fun! Now that we have our answer, let's make sure we understand what it represents and how it fits into the context of the original problem. In the next section, we'll put the final touches on our solution and present it in the required a + bi form. So, stick around – we're almost there!

Expressing the Result in Simplest a + bi Form

Alright, team, we've done the heavy lifting! We've calculated that (f-g)(√-9) = (f-g)(3i) = -20 - 6i. Now, let's make sure our answer is presented in the simplest a + bi form, just like the problem asked. This is the final polish on our masterpiece, guys. The beauty of our solution is that it's already in a + bi form! Remember, a + bi is the standard way to write a complex number, where a is the real part and b is the imaginary part. In our case, we have: -20 - 6i. Here, a = -20 and b = -6. There's nothing more we need to simplify or rearrange. Our answer is perfectly poised in the required format. Sometimes, you might encounter complex numbers that need a bit of tidying up before they're in the simplest a + bi form. This could involve combining like terms, rationalizing denominators, or other algebraic manipulations. But in our case, the calculation led us directly to the desired form. This final step is a reminder of the importance of paying attention to the instructions. The problem specifically asked for the answer in simplest a + bi form, and we've made sure to deliver exactly that. It's like adding the perfect frame to a painting – it completes the presentation. Presenting your answer correctly is as crucial as getting the math right. It shows that you understand the conventions and expectations of mathematical notation. So, always double-check that your final answer matches the format requested in the problem. Now that we've successfully expressed our result in simplest a + bi form, we can confidently say that we've solved the problem! We've navigated through function subtraction, imaginary numbers, and complex number representation. We've demonstrated a solid understanding of the concepts and the ability to apply them effectively. But more than just getting the answer, we've also gained a deeper appreciation for the process of mathematical problem-solving. We've learned how to break down complex problems into smaller, manageable steps, how to use definitions and properties to our advantage, and how to present our solutions clearly and concisely. So, let's take a moment to celebrate our accomplishment! We've tackled a challenging problem and emerged victorious. And with the skills and knowledge we've gained, we're ready to take on even more mathematical adventures. Great job, guys!

Conclusion

Wow, guys, we did it! We successfully evaluated (f-g)(√-9), where f(x) = x² + x and g(x) = 3x + 11, and expressed the answer in the simplest a + bi form. Our final answer is -20 - 6i. Let's take a moment to recap the journey we took to get here. We started by understanding the individual functions, f(x) and g(x), and what they represent. We then tackled the concept of function subtraction, defining (f-g)(x) as f(x) - g(x) and simplifying it to x² - 2x - 11. Next, we encountered the square root of a negative number, √-9, which led us to the world of imaginary numbers. We learned that √-9 is equivalent to 3i, where i is the imaginary unit defined as √-1. We then plugged 3i into our (f-g)(x) function, carefully substituting and simplifying using the property i² = -1. This yielded the complex number -20 - 6i. Finally, we recognized that our answer was already in the simplest a + bi form, where a is the real part and b is the imaginary part. Throughout this process, we emphasized the importance of breaking down complex problems into smaller, manageable steps. We also highlighted the significance of understanding the underlying concepts and properties, such as function definitions, imaginary numbers, and complex number representation. By mastering these fundamentals, we were able to confidently navigate through the problem and arrive at the correct solution. More than just getting the right answer, we've also developed valuable problem-solving skills that will serve us well in future mathematical endeavors. We've learned how to approach challenging problems with a systematic and logical mindset. We've also gained a deeper appreciation for the beauty and elegance of mathematics. So, what's the takeaway from all of this? Well, it's that math can be challenging, but it's also incredibly rewarding. With the right approach and a solid understanding of the fundamentals, we can tackle even the most daunting problems. And, most importantly, we can have fun along the way! So, keep exploring, keep learning, and keep challenging yourself. The world of mathematics is vast and fascinating, and there's always something new to discover. Thanks for joining me on this mathematical adventure, and I hope you found it both informative and enjoyable. Until next time, keep those mathematical gears turning!