Finding F(i) For F(x) = X³ - 2x²: A Step-by-Step Guide

Are you grappling with complex numbers and polynomial functions? This article provides a comprehensive, step-by-step guide on how to find f(i) when given the function f(x) = x³ - 2x². We'll break down the process, ensuring you understand each step, from substituting i into the function to simplifying the expression using the properties of imaginary units. By the end of this guide, you'll be able to confidently tackle similar problems involving complex numbers and polynomial functions. Let's dive in and unravel the mysteries of complex number evaluation in polynomial functions!

Understanding the Problem

Before we delve into the solution, let's clearly understand the problem. We are given a function, f(x) = x³ - 2x², which is a polynomial function. Our task is to find the value of this function when x is equal to i, where i represents the imaginary unit. In the realm of mathematics, the imaginary unit i is defined as the square root of -1 (i = √-1). This means that i² = -1, a crucial property we'll use later. Substituting i into the function essentially means replacing every instance of x in the expression x³ - 2x² with i. This will result in an expression involving powers of i, which we'll then simplify using the properties of imaginary units to arrive at our final answer. This process combines algebraic substitution with the unique characteristics of complex numbers, providing a solid exercise in mathematical manipulation and understanding. So, let's embark on this journey of substitution and simplification to uncover the value of f(i).

Step 1: Substitute i into the Function

The first step in solving this problem is to directly substitute i for x in the function f(x) = x³ - 2x². This is a straightforward application of function evaluation, where we replace the variable x with the given value, which in this case is the imaginary unit i. When we perform this substitution, we obtain the expression f(i) = (i)³ - 2(i)². Notice how every instance of x has been replaced by i. This substitution sets the stage for the next step, where we will simplify the powers of i and perform the necessary arithmetic operations. By replacing x with i, we've transformed the function evaluation problem into a problem of simplifying an expression involving complex numbers. This initial substitution is the cornerstone of the solution, allowing us to progress towards finding the equivalent expression for f(i). Now, let's move on to the crucial task of simplifying the powers of i.

Step 2: Simplify Powers of i

Now that we have f(i) = i³ - 2i², the next critical step is to simplify the powers of i. This involves understanding the cyclic nature of imaginary unit powers. We know that i = √-1, i² = -1, i³ = i² * i* = -1 * i = -i, and i⁴ = i² * i² = (-1) * (-1) = 1. This pattern repeats for higher powers of i. Therefore, to simplify i³, we can rewrite it as -i. The term i² is simply equal to -1. Substituting these simplified values back into our expression, we get f(i) = -i - 2(-1). This simplification is crucial because it allows us to express the function evaluation in terms of real and imaginary components, paving the way for further simplification and ultimately the final answer. By leveraging the fundamental properties of imaginary unit powers, we've reduced the complexity of the expression and brought ourselves closer to the solution.

Step 3: Perform Arithmetic Operations

With the powers of i simplified, we now have the expression f(i) = -i - 2(-1). The next step involves performing the arithmetic operations to further simplify this expression. First, we multiply -2 by -1, which gives us +2. Our expression then becomes f(i) = -i + 2. This is a straightforward arithmetic operation that combines the real and imaginary components of the expression. Now, rearranging the terms to present the real part first, we can write f(i) = 2 - i. This rearrangement is conventional when dealing with complex numbers, as it clearly separates the real and imaginary parts in the standard form a + bi, where a is the real part and b is the imaginary part. By performing this simple multiplication and rearranging the terms, we've arrived at a much cleaner and more easily interpretable form of the expression for f(i). This simplified form directly leads us to the final answer, which we will discuss in the next section.

Step 4: Identify the Equivalent Expression

After simplifying the expression, we have found that f(i) = 2 - i. Now, we need to identify which of the given options matches this result. Looking back at the original problem, we typically have multiple-choice options. The correct answer is the option that exactly matches our simplified expression, which is 2 - i. This step is a crucial verification process to ensure that all the previous steps have been performed correctly and that we have arrived at the correct solution. By comparing our simplified expression with the given options, we can confidently select the equivalent expression for f(i). In this case, the correct answer is the option that represents the complex number 2 - i, where 2 is the real part and -1 is the coefficient of the imaginary part. This final step solidifies our understanding of the problem and confirms our ability to evaluate polynomial functions with complex number inputs.

Conclusion

In this comprehensive guide, we've successfully navigated the process of finding f(i) for the function f(x) = x³ - 2x². We began by understanding the problem and recognizing the need to substitute the imaginary unit i into the function. We then meticulously substituted i for x, simplified the powers of i using their cyclic properties, performed the necessary arithmetic operations, and finally identified the equivalent expression for f(i). The key takeaways from this exercise include the importance of understanding the properties of imaginary units (i² = -1) and the ability to perform algebraic manipulations with complex numbers. This process not only provides the solution to this specific problem but also equips you with the skills to tackle similar problems involving polynomial functions and complex number evaluations. Remember, practice is key to mastering these concepts, so keep exploring and challenging yourself with different problems. For further exploration of complex numbers and their properties, you might find the resources at Khan Academy's Complex Numbers helpful. This resource offers a wealth of information and practice exercises to enhance your understanding.