Problem 44 - College Algebra

Question

Express the given function h as a composition of two functions f and g so that \(h(x)=(f \circ g)(x)\) $$h(x)=|3 x-4|$$

Step-by-Step Solution

Answer

The function \(h(x) = |3x - 4|\) can be expressed as a composition: \(h(x) = (f \circ g)(x)\), where \(f(x) = |x|\) and \(g(x) = 3x - 4\).

Step 1: Identify the operations

The function \(h(x)\) is executing two operations: 1) Multiplying \(x\) by 3 and subtracting 4 (3x - 4), 2) Taking the absolute value. The function \(g(x)\) can be the first operation, and the function \(f(x)\) can be the second operation.

Step 2: Define the two functions

Let's define the function \(g(x) = 3x - 4\). This is the first operation happening inside the absolute value operation. Then, let's define function \(f(x) = |x|\). This function is taking the absolute value of whatever argument we pass it.

Step 3: Make sure the composition matches the original function

The composition of the functions is \(f(g(x)) = |3x - 4|\). This is the same as the original function \(h(x)\), so our functions \(f\) and \(g\) form the function \(h\) as a composition.

Key Concepts

Absolute Value

The absolute value of a number is a fundamental concept in algebra that represents the distance of that number from zero on a number line. This means no matter whether the number is positive or negative, its absolute value is always non-negative.
For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5. Mathematically, it is denoted using vertical bars, like this:

• \(|x|\)

The absolute value function transforms the variable inside it to be non-negative, making it crucial in many areas such as calculus and algebra when dealing with real-world distances or magnitudes. For expressions like \(|3x - 4|\), it ensures the entire expression inside becomes non-negative, which is particularly useful when solving equations where the sign must be considered.

Algebraic Functions

Algebraic functions are the building blocks of almost all mathematical expressions that appear in algebra. They consist of polynomial, rational, or root functions, which involve basic arithmetic operations like addition, subtraction, multiplication, division, and taking roots. These are expressed as equations with one or more variables.
A simple algebraic function could be linear, such as

• \(f(x) = 3x - 4\)

This function is a linear algebraic function because it represents a line when graphed. Here, 3 is the coefficient of the variable \(x\) and -4 is the constant term.
Algebraic functions can become complex, comprising multiple terms and operations, but they retain the essence of mathematical operations that define how inputs are transformed into outputs.

Composite Functions

Composite functions involve combining two or more functions to form a new function. This is represented by the notation

• \((f \circ g)(x) = f(g(x))\)

Such functions enable us to build complex operations step by step.
Taking the given problem, we are interested in the composite function

• \(h(x) = |3x - 4|\)

Here, we've constructed \(h(x)\) by combining two functions:

• \(g(x) = 3x - 4\)
• \(f(x) = |x|\)

The key benefit of composition is that it simplifies problems by breaking down intricate operations into manageable individual functions. In practical terms, composite functions allow us to model real-world processes efficiently by applying multiple transformations sequentially, each function performing part of the overall transformation process.