Solving F(i): A Step-by-Step Guide For Math Enthusiasts

Hey math lovers! Ever stumbled upon a problem that seems a bit… imaginary? Well, today we're diving into just that! We're gonna figure out how to solve for f(i) when given the function f(x) = x³ - 2x². Don’t worry; it's not as scary as it looks. We'll break it down into easy-to-digest steps. By the end, you'll be comfortable with complex numbers and how to manipulate them. So, let’s get started and unravel this mathematical mystery! We will explore the function and understand the given problem, then go through the step-by-step calculation to find the correct answer, and end with the final answer with explanations.

Understanding the Problem: The World of Complex Numbers

Alright, before we jump into the calculation, let's make sure we're all on the same page. The problem asks us to find the value of the function f(x) when x is i. But what exactly is i? Here comes the cool part – i represents the imaginary unit in mathematics. It's defined as the square root of -1. Yes, you read that right! The square root of a negative number. This concept expands the number system beyond real numbers, allowing us to deal with things that don't exist on the typical number line. Pretty wild, huh?

So, when we see f(i), it means we need to substitute every instance of x in the function with i. Our function is f(x) = x³ - 2x². Therefore, to find f(i), we replace x with i: f(i) = i³ - 2i². Now, our task is to simplify this expression. This involves understanding powers of i. Let's quickly refresh our memory on how i behaves when raised to different powers:

• i¹ = i
• i² = -1
• i³ = -i (because i³ = i² * i = -1 * i = -i)
• i⁴ = 1 (because i⁴ = i² * i² = -1 * -1 = 1)

Knowing these basic properties of i is crucial. Keep in mind that the powers of i cycle through these four values. This knowledge will be super handy as we proceed. Don't worry, we'll go step by step, and it'll all come together soon enough! Let’s proceed to the actual calculation!

Step-by-Step Calculation: Finding f(i)

Now, let's get down to the actual calculation of f(i). We've established that f(i) = i³ - 2i². Let's break this down:

1. Calculate i³: As we mentioned earlier, i³ = -i. So, we can replace i³ with -i in our equation.
2. Calculate i²: We know that i² = -1. Let’s substitute this into our equation as well.
3. Substitute and Simplify: Now, our equation looks like this: f(i) = -i - 2(-1). See how we're simplifying step by step? Great!
4. Further Simplification: Let's simplify – 2(-1). That becomes +2. So, our equation is now f(i) = -i + 2.
5. Rearrange: Finally, let’s rearrange the terms to make it look neater. We can rewrite the expression as f(i) = 2 - i.

And there you have it, folks! We have simplified the expression and found our answer. It wasn’t that bad, was it? We've successfully calculated f(i), which is a complex number, by breaking it down into manageable steps. Now, let’s check the possible answers to make sure we got the right one!

Finding the Correct Answer: Matching the Solution

Now that we have computed f(i) to be 2 - i, it's time to check which of the provided options matches our answer. Remember, the options were:

A. -2 + i B. -2 - i C. 2 + i D. 2 - i

By comparing our result (2 - i) with the options, it’s clear that option D. 2 - i is the correct one. Congratulations, you’ve correctly solved the problem! You have now mastered how to calculate the function of a complex number by understanding the properties of i and performing algebraic manipulations. You have successfully navigated the world of complex numbers and arrived at the correct solution. Isn't it satisfying when everything clicks? This journey shows that complex numbers, though initially intimidating, become approachable with a step-by-step strategy. This skill will be useful in more advanced math and physics. So, keep practicing and expanding your knowledge.

Conclusion: Wrapping It Up and Further Exploration

Awesome work, everyone! We've successfully calculated f(i), transforming a seemingly complex problem into a series of simple, manageable steps. We've understood the concept of imaginary numbers, the powers of i, and how to apply them to solve the given function. Remember, the key is breaking down the problem, understanding the fundamentals, and taking it one step at a time. The properties of i will come in handy as you advance in mathematics, especially in areas like algebra, calculus, and even in fields like electrical engineering and quantum physics!

Key Takeaways: This problem highlights the importance of understanding the basics. The ability to correctly solve this type of problem opens up the door to more intricate mathematical concepts. Here are some key points to remember:

• Imaginary Unit: Always remember what i represents.
• Powers of i: Know the cyclical pattern of i, i², i³, and i⁴.
• Step-by-Step Approach: Break down complex problems into smaller, manageable steps.

Feel free to explore other problems involving complex numbers and functions. Practice is key! You can try different functions and different values to get more practice. The more you practice, the more comfortable you will become with these concepts. Keep exploring, keep learning, and most importantly, keep enjoying the fascinating world of mathematics! If you have any further questions or want to explore other topics, just let me know. Happy calculating, and see you in the next math adventure!