Question

Given the complex function \(f(x)=(8+3 i)+(7-2 i) x\) of the real variable \(x\), what is \(|f(x)|^{2}\) ?

Short answer

Answer: The square of the magnitude of the complex function \(f(x)\) is given by \(|f(x)|^2=((8+7x)^2+(3-2x)^2)\).

Step-by-step solution

Write down the function

The complex function is given by \(f(x) = (8+3i) + (7-2i)x\). Let's rewrite the function as: \(f(x) = 8 + 3i + 7x - 2xi\).

Find the magnitude of \(f(x)\)

To find the magnitude of \(f(x)\), we can use the formula: \(|f(x)|=\sqrt{(\operatorname{Re}(f(x)))^2+(\operatorname{Im}(f(x)))^2}\), where \(\operatorname{Re}(f(x))\) and \(\operatorname{Im}(f(x))\) are the real and imaginary parts of \(f(x)\) respectively. In this case, the real part is \(\operatorname{Re}(f(x))=8+7x\), and the imaginary part is \(\operatorname{Im}(f(x))=3-2x\). Now, we can find the magnitude of \(f(x)\): \(|f(x)|=\sqrt{(8+7x)^2+(3-2x)^2}\).

Calculate the square of the magnitude

To find the square of the magnitude, we simply square the expression we got above: \(|f(x)|^2=((8+7x)^2+(3-2x)^2)\). This is our answer for the square of the magnitude of \(f(x)\) in terms of \(x\).

Key concepts

Complex Numbers

Complex numbers are an extension of the traditional real numbers and are essential in various fields of mathematics and engineering. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined as \(i^2 = -1\). The number \(a\) is known as the real part, while \(b\) is the imaginary part of the complex number.

Complex numbers can be visualized on a plane called the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This graphical interpretation is quite useful for understanding complex number operations such as addition, subtraction, multiplication, and division. When dealing with complex numbers, algebraic techniques can be applied much like they are with real numbers, but with special consideration to the properties of \(i\).

Magnitude Calculation

The magnitude (or modulus) of a complex number is a measure of its size, akin to how we consider the length of a vector. For a complex number \(z = a + bi\), the magnitude is denoted \(|z|\) and is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \).

When calculating the magnitude, we are essentially finding the distance from the origin (\(0,0\)) to the point \( (a, b) \) in the complex plane. For a complex function like \(f(x) = (8+3i) + (7-2i)x\), we treat the real and imaginary parts as functions of \(x\), which gives us a variable magnitude depending on the value of \(x\). By applying the magnitude calculation formula, we can determine the expression for the magnitude's square, \( |f(x)|^2 \[), and thus analyze the function's behavior for different real values of \(x\).

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. In the context of complex numbers, algebra involves operations that may include real and imaginary units. When simplifying expressions involving complex numbers, it is crucial to use algebraic rules such as the distributive law, the associative law, and the commutative law.

In the given exercise, algebra is used to combine like terms and to expand expressions to find the magnitude squared of the complex function. The process of squaring the magnitude, a concept rooted in algebraic manipulation, requires expanding the squares of binomials and combining like terms. A thorough understanding of algebra is essential to solving complex functions and their magnitudes, as seen in exercises like \( |f(x)|^2 = ((8+7x)^2 + (3-2x)^2) \).