Finding The Magnitude Of A Complex Function: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fascinating problem that blends algebra and complex numbers. Specifically, we'll figure out how to find the magnitude of a function when dealing with the imaginary unit, i. This is a fundamental concept, so let's break it down step by step to ensure everyone understands it perfectly.

Understanding the Question: If f(x) = 1 - x, What is the Value of |f(i)|?

So, what does this question really ask? We are given a function, f(x), defined as 1 - x. Our goal is to find the magnitude (or absolute value) of this function when the input is the imaginary unit, i. Let's first clarify what i is. In mathematics, i represents the imaginary unit, where i² = -1. This allows us to work with the square roots of negative numbers, expanding the realm of numbers beyond real numbers into complex numbers. Knowing that, let's substitute i into our function, which is the first important step, and then figure out the magnitude of the resulting complex number. Finding the value of |f(i)| involves substituting i into the function, which means replacing every instance of x in the function with i. The concept here is straightforward substitution. We simply replace x with i in the function f(x) = 1 - x. This gives us f(i) = 1 - i. The next part of the problem needs us to calculate the magnitude of the complex number 1 - i. Remember that complex numbers are expressed in the form a + bi, where a is the real part, and b is the imaginary part. In this case, our complex number is 1 - i, where a = 1 and b = -1. The magnitude, also known as the modulus, of a complex number a + bi is calculated using the formula |a + bi| = √(a² + b²). Using this, let's calculate the magnitude of 1 - i. With a = 1 and b = -1, we get |1 - i| = √(1² + (-1)²). Simplifying this, we get √(1 + 1), which equals √2. So, the value of |f(i)| is √2.

Now, let's explore some key concepts and then carefully analyze each option to determine the correct answer. The main keywords in this section are complex numbers, imaginary unit, and magnitude. The core of this problem lies in understanding complex numbers. Complex numbers extend real numbers by including an imaginary part, usually expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit, i, is defined as the square root of -1. This allows us to find the square root of negative numbers. The magnitude or modulus of a complex number a + bi is its distance from the origin (0,0) in the complex plane. It is calculated as √(a² + b²). The complex plane is a two-dimensional plane where the horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part. Each point on this plane corresponds to a complex number. To find |f(i)|, we first substitute i into the function f(x) = 1 - x, giving us f(i) = 1 - i. We then apply the magnitude formula: |1 - i| = √(1² + (-1)²) = √2. This gives us the correct answer.

Step-by-Step Solution: Evaluating |f(i)|

Let's break down the solution into smaller, manageable steps. This will help you understand the process better. First, we have the function f(x) = 1 - x. We need to substitute i for x. The main keyword here is substitution. By substituting i into the function, we get f(i) = 1 - i. Now that we have a complex number, we need to calculate its magnitude. Recall that the magnitude of a complex number a + bi is found using the formula √(a² + b²). In our case, the complex number is 1 - i, which can be written as 1 + (-1)i. The next important keyword is magnitude, so let's use it here. Here, a = 1 and b = -1. So, the magnitude |1 - i| = √(1² + (-1)²). Now let's calculate and simplify the equation. This simplifies to √(1 + 1), which equals √2. The question asks for the magnitude, which we have now calculated. Therefore, |f(i)| = √2. Understanding this process thoroughly is key to solving similar problems. Let’s walk through the steps methodically. First, start with the function definition: f(x) = 1 - x. The first step is to substitute x with i, to get f(i) = 1 - i. The second step is to find the magnitude of the complex number 1 - i. The real part, a, is 1, and the imaginary part, b, is -1. Using the magnitude formula, we calculate √(1² + (-1)²) = √2. So, the final answer is √2. The value is a non-negative real number, representing the distance from the origin in the complex plane.

Analyzing the Answer Choices

Now, let's look at the answer choices. Understanding each option and why it is right or wrong is just as important as solving the problem itself. Our main keyword here is evaluation, as we're going to evaluate each answer option. Analyzing the answer options is a critical step in problem-solving. It helps to solidify our understanding and ensure that we've correctly applied the concepts. The question asks for the value equivalent to |f(i)|. We know that |f(i)| = √2. Let's analyze each option.

A. 0: This is incorrect. 0 is a real number, but it is not the magnitude of 1 - i. B. 1: This is incorrect. 1 is a real number, but it's not the correct magnitude. It might come from misinterpreting parts of the calculation, but it is not the actual magnitude. C. √2: This is correct. This is the magnitude we calculated using the formula. It represents the distance of the complex number 1 - i from the origin in the complex plane. D. √-1: This is incorrect. √-1 is equivalent to i. Although i is involved in the calculation, it's not the answer, and it doesn't represent the magnitude. The correct answer must be a real, non-negative number.

So, after evaluating all options, we can clearly see the correct answer is C. √2. Recognizing the definition and method behind a complex number problem enables you to correctly navigate the answer choices.

Conclusion: Mastering Complex Number Magnitudes

Congratulations! We've successfully navigated the process of finding the magnitude of a complex function. By understanding the basics of complex numbers, the imaginary unit, and the magnitude formula, you've equipped yourself to solve similar problems confidently. Remember, practice is key. Try solving similar problems to reinforce your understanding. Always focus on breaking down complex problems into smaller, more manageable steps. Identify key concepts such as substitution, complex numbers, and magnitude. Regularly reviewing these concepts will significantly improve your skills.

By following this step-by-step guide and practicing regularly, you'll master this concept and be well-prepared to tackle any similar challenges. Keep exploring, keep practicing, and keep that mathematical curiosity alive!

For further learning on complex numbers and related concepts, consider checking out resources on Khan Academy and MathWorld. These websites provide detailed explanations and practice problems to help you build a stronger foundation in mathematics. Good luck, and keep exploring the fascinating world of mathematics!

Here's a link to a resource where you can learn more about complex numbers: Khan Academy - Complex Numbers