Problem 34 - Precalculus

Question

Write the given function as a composition of two or more non-identity functions. (There are several correct answers, so check your answer using function composition.) $$H(x)=|7-3 x|$$

Step-by-Step Solution

Answer

The function can be expressed as a composition: \( H(x) = f(g(x)) \) where \( g(x) = 7 - 3x \) and \( f(u) = |u| \).

Step 1: Identify the Main Components

The given function is \( H(x) = |7 - 3x| \). To express this function as a composition, observe that it consists of an absolute value operation and a linear transformation \( 7 - 3x \). We need to split this function into two separate functions, one for each operation.

Step 2: Define the Inner Function

The inner function performs the linear transformation. Let's define it as \( g(x) = 7 - 3x \). This function takes an input \( x \) and maps it to a new expression, \( 7 - 3x \), which will then become the input to the next function in our composition.

Step 3: Define the Outer Function

The outer function applies the absolute value operation. We define this function as \( f(u) = |u| \), where \( u \) is a placeholder for whatever input it receives from the inner function \( g(x) \). This function will take the result of \( g(x) \) and apply the absolute value.

Step 4: Compose the Functions

In function composition, the outer function \( f \) takes the result of the inner function \( g \). Thus, \( H(x) = f(g(x)) \). Substituting the definitions of the functions, we have \( f(g(x)) = f(7 - 3x) = |7 - 3x| \). This confirms the composition \( H(x) = f(g(x)) \) is valid.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It involves calculating the absolute value of a number, which is the distance of that number from zero on a number line. The absolute value of a number is always non-negative.
For example, the absolute value of -3 is 3, and the absolute value of 3 is also 3, because both are 3 units away from 0.
The mathematical notation for absolute value is two vertical bars surrounding the number or expression: \(|x|\).
It's important to understand that absolute value functions can affect any operations inside. For instance, \(|7 - 3x|\) implies an absolute value computation of the linear expression \(7 - 3x\), ensuring the result is always non-negative. This characteristic makes the absolute value function crucial in situations where the sign doesn't matter, only the magnitude or size of the number.

Linear Transformation

A linear transformation can be seen as a function that represents a line. In our example, the linear transformation is represented by the function \( g(x) = 7 - 3x \).
This specific transformation combines multiplication and subtraction to adjust the input, \(x\), by scaling it and then shifting it. Here's how it works:

• The term \(-3x\) shows that for each increase by 1 in \(x\), the result of the transformation decreases by 3. This is known as the slope of the line.
• The constant 7 is the y-intercept, meaning that when \(x = 0\), the function outputs 7. This value shows where the line crosses the y-axis.

Linear transformations are vital in understanding how inputs are altered geometrically, creating lines of various slopes and positions on a graph, based on their specific formulas.

Composite Functions

Composite functions work by nesting one function inside another. This is a way of combining two or more functions to create a new function, which is incredibly powerful in mathematics. When we say compose two functions, typically denoted as \( f(g(x)) \), it means the outcome of \( g(x) \) becomes the input for \( f(x) \).
In our exercise, we define two functions: the inner one \( g(x) = 7 - 3x \) and the outer one \( f(u) = |u| \).
When you compose these two functions, you first apply \( g \) to \( x \) and then apply \( f \) to the result of \( g \), resulting in \( H(x) = |7 - 3x| \). This showcases the elegance of composite functions, where multiple operations are seamlessly applied in sequence, highlighting the modularity and flexibility of function composition in mathematics.