Understanding Complex Numbers: The Absolute Value Of F(i)

Welcome, math enthusiasts! Today, we're diving into the fascinating world of complex numbers and exploring how to find the absolute value of a function evaluated at an imaginary number. Specifically, we'll tackle the question: If $f(x) = 1-x$, which value is equivalent to $|f(i)|$? This problem might look a little intimidating at first, especially with the introduction of '$i$', the imaginary unit. But don't worry, we'll break it down step-by-step, making sure you understand every part of the process. We'll demystify what '$f(i)$' means and how to calculate its absolute value, often referred to as its magnitude. By the end of this discussion, you'll be equipped to solve similar problems with confidence. We'll cover the definition of the imaginary unit, how to substitute it into a function, and the fundamental concept of the absolute value of a complex number. So, grab your thinking caps, and let's embark on this mathematical journey together!

The Foundation: Understanding the Imaginary Unit '$i$'

Before we can find the value of $|f(i)|$, we first need to get a solid grip on what the imaginary unit, represented by '$i$', actually is. In the realm of real numbers, we know that the square of any real number is always non-negative. For instance, $2^2 = 4$ and $(-3)^2 = 9$. This means there's no real number that, when squared, results in a negative number. However, mathematicians encountered situations where they needed to solve equations like $x^2 + 1 = 0$. If we try to solve this using real numbers, we'd get $x^2 = -1$, which has no real solution. To address this, the imaginary unit '$i$' was introduced. The fundamental definition of '$i$' is that it is the square root of -1, or $i^2 = -1$. This single definition unlocks an entirely new set of numbers called complex numbers. A complex number is generally expressed in the form $a + bi$, where '$a$' is the real part and '$b$' is the imaginary part. For example, $3 + 2i$ is a complex number with a real part of 3 and an imaginary part of 2. The imaginary unit '$i$' itself can be thought of as a complex number where the real part is 0 and the imaginary part is 1, i.e., $0 + 1i$. Understanding this basic property, $i = \sqrt{-1}$, is crucial because it's the cornerstone of all operations involving complex numbers, including the one we'll be performing today.

Evaluating the Function: Finding $f(i)$

Now that we're comfortable with the imaginary unit '$i$', let's move on to the next step: evaluating our given function, $f(x) = 1-x$, at $x=i$. This means we need to substitute '$i$' wherever we see '$x$' in the function's expression. So, $f(i) = 1 - i$. Congratulations! You've just performed your first function evaluation with a complex number. The result, $1-i$, is itself a complex number. Remember, a complex number has a real part and an imaginary part. In the case of $1-i$, the real part is 1 and the imaginary part is -1. It's important to note that this is not the final answer to our problem, but it's a critical intermediate step. We've successfully translated the abstract notation '$f(i)$' into a concrete complex number that we can now work with. This process is straightforward: identify the input value, and replace every instance of the variable in the function with that input value. Even if the input is a complex number like '$i$', the substitution process remains the same. This ability to substitute and evaluate functions with complex inputs is a fundamental skill in complex analysis and many areas of science and engineering.

The Magnitude of a Complex Number: Understanding $|f(i)|$

Our final step is to find the absolute value, or magnitude, of the complex number we just obtained, $f(i) = 1-i$. The absolute value of a complex number $z = a + bi$ is denoted as $|z|$ and is calculated using the Pythagorean theorem. It represents the distance of the complex number from the origin (0,0) in the complex plane. The formula for the absolute value is $|z| = \sqrt{a^2 + b^2}$. In our case, the complex number is $f(i) = 1 - i$. Here, the real part '$a$' is 1, and the imaginary part '$b$' is -1. Applying the formula, we get $|f(i)| = |1 - i| = \sqrt{1^2 + (-1)^2}$. Let's calculate this: $1^2 = 1$ and $(-1)^2 = 1$. So, $|f(i)| = \sqrt{1 + 1} = \sqrt{2}$. The absolute value of $f(i)$ is $\sqrt{2}$. This value represents the distance from the origin to the point (1, -1) in the complex plane. It's a real, non-negative number, as expected for an absolute value. Understanding the absolute value is key because it gives us a sense of the 'size' or 'magnitude' of a complex number, irrespective of its direction in the complex plane.

Conclusion: Solving $|f(i)|$ for $f(x)=1-x$

We have successfully navigated the process of finding the value equivalent to $|f(i)|$ when $f(x) = 1-x$. By first understanding the imaginary unit '$i$' as the square root of -1, we proceeded to substitute '$i$' into the function to get $f(i) = 1 - i$. Finally, we applied the formula for the absolute value of a complex number, $|a + bi| = \sqrt{a^2 + b^2}$, to our result. For $1-i$, where $a=1$ and $b=-1$, the absolute value is $|1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$. Therefore, the value equivalent to $|f(i)|$ is $\sqrt{2}$. This aligns with option C in the multiple-choice selection. This problem beautifully illustrates how basic algebraic functions can be extended to the complex number system, and how concepts like absolute value take on a geometric interpretation in the complex plane. The journey from a simple linear function to calculating the magnitude of a complex number highlights the elegance and power of mathematics.

For further exploration into the fascinating world of complex numbers and their applications, I recommend visiting the Khan Academy website. They offer a wealth of resources and tutorials that can deepen your understanding of these concepts.