Problem 37 - 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.) $$q(x)=\frac{|x|+1}{|x|-1}$$

Step-by-Step Solution

Answer

The function can be expressed as a composition: \( q(x) = h(g(x)) \) with \( g(x) = |x| \) and \( h(u) = \frac{u+1}{u-1} \).

Step 1: Identify Inner Function

We begin by breaking down the function into parts and identifying a potential inner function. Notice that the expression \( |x| \) appears in both the numerator and the denominator. Let's define \( g(x) = |x| \). This transforms \( q(x) \) into \( q(x) = \frac{g(x) + 1}{g(x) - 1} \).

Step 2: Define Outer Function

Next, recognize the entire fraction as another function based on our new expression in terms of \( g(x) \). Let's define a new function \( h(u) = \frac{u + 1}{u - 1} \). Therefore, the entire function \( q(x) \) can be expressed as \( q(x) = h(g(x)) \), where \( g(x) = |x| \) and \( h(u) = \frac{u + 1}{u - 1} \).

Key Concepts

Understanding Inner Functions

In the realm of function composition, the inner function is a crucial part of understanding how a composed function operates. The inner function represents the initial processing or transformation of an input variable that lays the groundwork for further manipulation by the outer function. In simple terms, the inner function is the first function applied in a sequence of functions.

• The inner function is usually denoted as \( g(x) \) in the composition \( (f \circ g)(x) \), where \( f \) is the outer function.
• Its output serves as the input for the next function in the chain.

By breaking down a complicated function into smaller parts, such as identifying the inner function, you can simplify the task of analyzing or computing the overall function. This is particularly useful in cases where the inner function represents a simple transformation, such as the absolute value or a polynomial. The clearer definition of each part enhances our understanding and ensures any transformations are implemented correctly.

Identifying the Outer Function

The outer function in a composition takes the result from the inner function and applies further operations. Think of it as the finishing touch to complete the overall transformation.

• The outer function is often represented as \( h(u) \) or \( f(x) \) in a function pair \( (f \circ g)(x) \).
• It is crucial because it defines the ultimate result of the composition once the inner function's result is passed on to it.

For instance, in the given exercise, after defining the inner function \( g(x) = |x| \), the fraction \( h(u) = \frac{u+1}{u-1} \) represents the outer function. This function receives the processed input via \( g(x) \) and performs a final operation to provide the complete picture of the function \( q(x) \). Understanding each function's role allows us to see how the values are shifted and scaled, making complex functions manageable.

Exploring the Absolute Value Function

The absolute value function is a fundamental mathematical concept often used in function compositions due to its unique properties. In function notation, it is expressed as \( |x| \), and its primary role is to return the non-negative value of \( x \), regardless of whether \( x \) itself is positive or negative.

• If \( x \geq 0 \), then \(|x| = x\).
• Conversely, if \( x < 0 \), then \(|x| = -x\).
• This results in \(|x|\) always being \( \geq 0 \).

When utilized as the inner function in a composition, such as \( g(x) = |x| \), it serves to normalize or reduce the input to a standard form before further transformations. This can be particularly beneficial in scenarios that require the analysis of magnitude without directional influence, simplifying complex equations and ensuring the combined operations maintain specific properties. This characteristic makes the absolute value function an invaluable tool in mathematical problem-solving.