Problem 80 - Precalculus Essentials

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 \(h\) can be decomposed into two functions \(f\) and \(g\) such that \(f(x) = |x|\) and \(g(x) = 3x - 4\), effectively making \(h(x) = (f \circ g)(x) = |3x - 4|\).

Step 1: Identify the inner function

Looking at the function \(h(x)=|3 x-4|\), the first operation that is applied to \(x\) is the multiplication by 3 and the subtraction of 4. This inner operation can act as function \(g(x)\). As a result, we can say that \(g(x) = 3x - 4\).

Step 2: Identify the outer function

With \(g(x)\) defined, the next step is to identify the outer function. This is the function that is applied to the outcome of \(g(x)\). In this case, this function is the absolute value function \(|x|\). Thus we define \(f(x) = |x|\).

Step 3: Verify the decomposition

To ensure that the decomposition is correct, substitute \(g(x)\) into \(f\) and see if you re-construct \(h(x)\). So \(h(x) = f(g(x)) = f(3x - 4) = |3x - 4|\). Since this is the same as the original function, the decomposition of \(h\) into \(f\) and \(g\) is correct.

Key Concepts

Inner and Outer Functions

Understanding the concept of inner and outer functions is like peeling an onion—there's a process to which part you access first. In mathematical terms, when we compose two functions, the function we apply first is called the 'inner function,' while the function applied afterward is known as the 'outer function.' This decomposition into two separate entities allows us to comprehend complex functions by breaking them down into simpler parts.

For instance, if we want to express a function such as \( h(x) = |3x - 4| \) as a composition, we need to look at the order in which we perform the operations. The inner function is the one closest to the variable \( x \), here it would be the linear operation \( g(x) = 3x - 4 \). This inner function is then fed into the outer function \( f \), which, in this example, is the absolute value function. Hence, the outer function is definitely \( f(x) = |x| \). The composition \( (f \(circ\) g)(x) \) reflects the layered application of these functions.

Absolute Value Function

The absolute value function, denoted as \( |x| \), is a fundamental concept in math which relates to distance. Specifically, it's the distance a number is from zero on the number line, without considering direction. This function is pivotal in various fields such as engineering, physics, and economics because it expresses magnitude without regard to sign.

The graph of the absolute value function is a V-shape, indicating that all negative inputs become positive outputs after the function is applied. In practice, when we decompose more complex functions and identify an absolute value function as the outer layer, it implies that the result will always be non-negative. For example, if we have \( h(x) = |3x - 4| \), applying \( g(x) = 3x - 4 \) first yields a value that may be negative or positive, but once we apply the outer layer—\( f(x) = |x| \), the final result is guaranteed to be positive or zero.

Function Operations

Function operations are like tools in a mathematician's toolkit, allowing you to build new functions from existing ones. These operations include addition, subtraction, multiplication, division, and composition of functions. Each of these operations has rules and properties that need to be understood for the proper manipulation of functions.

In the context of our example function \( h(x) = |3x - 4| \), the operation involved is function composition, denoted by \( (f \(circ\) g)(x) \). This composition means evaluating the inner function,\( g(x) \), first and then applying the outer function,\( f \), to the result. The outcome produces a new function with properties derived from both the constituents—\( f \) and \( g \). Function composition is a vital concept in calculus and higher-level mathematics as it paves the way to understanding more complicated functions and their behaviors under various operations.