Problem 32 - Algebra A Combined Function

Question

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=|x-1| $$

Step-by-Step Solution

Answer

\( f(x) = |x| \) and \( g(x) = x-1 \).

Step 1: Choose a Function for g(x)

To decompose the function \( h(x) = |x - 1| \), start by choosing a simple function for \( g(x) \). Let's set \( g(x) = x - 1 \). This choice helps us transform \( x \) before applying the absolute value operation.

Step 2: Determine f(x) based on g(x)

Given \( g(x) = x - 1 \), we notice that \( h(x) \) requires the operation \( |x| \). This can be achieved by defining \( f(x) = |x| \). Thus, after substituting \( g(x) \) into \( f(x) \), we will have \( f(g(x)) = |x - 1| \), which matches \( h(x) \).

Step 3: Confirm the Composition

Using the functions \( f(x) = |x| \) and \( g(x) = x - 1 \), verify the composition \( (f \circ g)(x) = f(g(x)) \). Substitute \( g(x) \) into \( f(x) \) to confirm: \[ f(g(x)) = f(x - 1) = |x - 1| \]. This confirms \( (f \circ g)(x) \) correctly reproduces \( h(x) \).

Key Concepts

composition of functions

Composition of functions is an important mathematical concept that involves using the output of one function as the input for another.
In this exercise, the function composition is symbolized by \( (f \circ g)(x) \).
The notation \( f \circ g \) represents a composite function, meaning that you first apply the function \( g(x) \) and then apply the function \( f(x) \) to the result.

Consider the following characteristics:

• Order matters: Apply the functions in the correct sequence. Here, \( g(x) \) comes first, followed by \( f(x) \).
• The domain of the composition is determined by both functions: It's the range of \( g(x) \) that feeds into \( f(x) \).

In the solution, we defined: \( g(x) = x - 1 \) and \( f(x) = |x| \).
This results in \( (f \circ g)(x) = |x - 1| \), matching the given function \( h(x) \). By decomposing the function this way, you can often reveal the underlying operations that make up a more complex function.

absolute value function

The absolute value function is a fundamental tool in mathematics, often denoted as \( |x| \).
This function extracts the magnitude of a number, ignoring its sign.

Key properties of the absolute value function include:

• For any real number \( x \), \( |x| = x \) if \( x \geq 0 \), and \( |x| = -x \) if \( x < 0 \).
• This absolute value transformation ensures non-negative results.

In the context of our exercise, the absolute value function is applied after subtracting 1 from \( x \), through the function \( f(x) = |x| \). This application helps in measuring the distance of any number to 1 along the real number line, without considering the direction. It's an effective way to ensure that the output of our composed function \( |x-1| \) gives a non-negative value irrespective of the input.

function transformation

Function transformation involves modifying a standard function to change its graph's position or shape.
This concept is crucial when working on problems involving function decomposition.

Various types of transformations include:

• Translation: Shifting the function's graph horizontally or vertically. For example, \( x - 1 \) shifts the graph of \( x \) one unit to the right.
• Reflection: Flipping the graph. The absolute value can imply a reflection if applied inside or outside of the parentheses, turning \( x \) into \( |x| \).
• Dilation: Stretching or compressing the graph vertically or horizontally.

In our composed function \( h(x) = |x-1| \), the transformation is primarily a horizontal shift.
The function \( g(x) = x - 1 \) moves the basic \( |x| \) graph one unit to the right.
This shift is crucial for understanding how more complicated functions can derive from simple transformations of elementary functions.

domain and range

Understanding the domain and range of a function is vital for analyzing its behavior and potential output.
The domain refers to all the possible input values (or \( x \) values) of the function, while the range refers to all possible output values (or \( y \) values).

Consider the following aspects:

• Domain: For \( h(x) = |x-1| \), \( x \) can be any real number, making its domain \((-\infty, \infty)\)
• Range: Because absolute value functions are non-negative, the range of this function is \([0, \infty)\).

When decomposing functions, it's crucial to confirm that the range of \( g(x) \) is compatible with the domain of \( f(x) \).
In our exercise, since \( f(x) = |x| \) accepts any real number and outputs non-negative numbers, the composition operates smoothly.
The analyzed domain and range are helpful to predict the graph behavior and ensure function validity across the chosen values.