Problem 6 - Precalculus

Question

Evaluate the algebraic expressions. If \(f(x)=x^{3}-2,\) evaluate \(f(i)\)

Step-by-Step Solution

Answer

The value of \(f(i)\) is \(-i - 2\).

Step 1: Substitute the Value

First, we need to substitute the given value, which is the imaginary unit \(i\), into the function \(f(x) = x^3 - 2\). So, replace \(x\) with \(i\): \[ f(i) = i^3 - 2 \]

Step 2: Simplify the Expression

Next, simplify the expression \(i^3 - 2\). Using the property of imaginary numbers, \(i^2 = -1\), we find: \[ i^3 = i^2 imes i = -1 imes i = -i \]Thus, the expression becomes: \[ f(i) = -i - 2 \]

Key Concepts

Complex Numbers

Complex numbers are numbers that have both real and imaginary parts. They are usually written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. In this notation, \(i\) is the imaginary unit, which we will learn more about in the next section.
\[ \]Complex numbers can be visualized as points or vectors in a two-dimensional plane, known as the complex plane. The horizontal axis represents the real part, while the vertical axis represents the imaginary part.
\[ \]

• For instance, the complex number \(3 + 4i\) has 3 as its real part and 4 as its imaginary part.


• You can think about it as a point located "+3" in the real direction and "+4" in the imaginary direction.

The addition and subtraction of complex numbers can be done by separately adding or subtracting their real and imaginary components. Multiplication involves distributing like factor terms and using the property that \(i^2 = -1\). Remember, working with complex numbers is much like working with binomials, just with an added twist for the imaginary part.

Imaginary Unit

The imaginary unit, represented by \(i\), is a mathematical concept that is used to extend the real numbers to complex numbers. The core property that defines \(i\) is:
\[ \]- \(i^2 = -1\)
\[ \]Using this property, we can derive other powers of \(i\):
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• \(i^3 = i^2 \times i = -1 \times i = -i\)
• \(i^4 = (i^2)^2 = (-1)^2 = 1\)

After \(i^4\), the powers of \(i\) begin to repeat: \(i^5 = i\), \(i^6 = -1\), and so on. This cyclical nature greatly aids in simplifying expressions involving powers of \(i\).
\[ \]When working with functions, substituting \(i\) allows us to explore how imaginary components affect the result. This is precisely what happens when evaluating functions at complex numbers.

Function Evaluation

When evaluating functions like \(f(x) = x^3 - 2\), we substitute the input value, in this case, the imaginary unit \(i\), into the function. This process transforms the function into an algebraic expression that we can simplify using known rules of arithmetic for both real and imaginary calculations.
\[ \]1. **Substitute the given value:** In the function \(f(x) = x^3 - 2\), replace \(x\) with \(i\) which gives us \(f(i) = i^3 - 2\).
2. **Simplify the expression:** Use the properties of \(i\). We've already established \(i^3 = -i\), hence \(f(i) = -i - 2\).
\[ \]These steps show how you can evaluate an expression even when complex numbers are involved. It's a matter of understanding the properties of imaginary numbers and applying them accordingly. This is a critical skill in algebra, especially as you begin to tackle more advanced problems.