To solve the given problem, we need to first evaluate the function f(x) at i. Recall that i is the imaginary unit, which is defined as the square root of -1. Therefore, we have:
f(i) = 1 – i
Next, we need to find the absolute value of f(i). The absolute value of a complex number is defined as the distance between the origin and the complex number on the complex plane. To do this, we can use the Pythagorean theorem, which states that the magnitude of a complex number can be calculated by taking the square root of the sum of the squares of its real and imaginary components.
So, let’s calculate the absolute value of f(i):
|f(i)| = √(1^2 + (-1)^2) = √(1 + 1) = √2
Therefore, the value equivalent to |f(i)| is √2.
Now, let’s explore some additional concepts related to complex numbers and absolute values.
Complex numbers are numbers that consist of both a real component and an imaginary component. They can be represented on a two-dimensional plane called the complex plane, where the horizontal axis represents the real component and the vertical axis represents the imaginary component.
The absolute value of a complex number z can be written as |z|. It is also sometimes referred to as the modulus or magnitude of the complex number. The absolute value of a complex number is always a non-negative real number. Geometrically, it represents the distance between the origin and the point representing the complex number on the complex plane.
The formula for calculating the absolute value of a complex number z = x + iy is:
|z| = √(x^2 + y^2)
where x is the real part of the complex number and y is the imaginary part.
The absolute value of a complex number has many useful properties. For example, the absolute value of the product of two complex numbers is equal to the product of their absolute values:
|zw| = |z||w|
where z and w are any two complex numbers.
Another important property of the absolute value of a complex number is that it satisfies the triangle inequality:
|z + w| ≤ |z| + |w|
where z and w are any two complex numbers. This inequality states that the absolute value of the sum of two complex numbers cannot be greater than the sum of their absolute values.
In conclusion, the value equivalent to |f(i)| in the given problem was √2. The absolute value of a complex number is an important concept in mathematics and has many useful properties. It represents the distance between the origin and the point representing the complex number on the complex plane and can be calculated using the Pythagorean theorem.
