Problem 53 - Algebra and Trigonometry

Question

Express the function in the form \(f \circ g\) $$ H(x)=\left|1-x^{3}\right| $$

Step-by-Step Solution

Answer

\(H(x) = f(g(x))\) where \(f(u) = |u|\) and \(g(x) = 1 - x^3\).

Step 1: Identify the Inner Function

Look for a function that is encapsulated inside an outer function. In this case, notice that the expression \(1 - x^3\) is inside the absolute value function. Let's define this as the inner function \(g\). So, \(g(x) = 1 - x^3\).

Step 2: Define the Outer Function

The entire function \(H(x) = \left|1 - x^3\right|\) is essentially taking the absolute value of the expression \(1 - x^3\). Hence, the outer function \(f\) involves taking the absolute value. Define this as \(f(u) = |u|\), where \(u\) represents any expression.

Step 3: Construct the Composite Function

Using the definitions from the previous steps, the composite function \(f \circ g\) is \(f(g(x)) = f(1 - x^3)\). Substituting \(g(x)\) into \(f\), we have \(f(1 - x^3) = |1 - x^3|\), which matches the original function \(H(x)\).

Key Concepts

Inner Function

In the realm of composite functions, identifying the 'Inner Function' is the first step. Imagine it as the core or the heart of your expression. In our exercise, the expression within the absolute value function, \(1 - x^3\), stands out. This is because it's enclosed—like filling inside a sandwich—by the absolute value. Our inner function, which we'll call \(g(x)\), is this very part: \(g(x) = 1 - x^3\). Learn more about inner functions here

Outer Function

Once you've grasped the heart of the expression, we move to the outside shell—the 'Outer Function'. This is the function that acts on the result of your inner function. For our problem, the operation wrapping around \(1 - x^3\) is the absolute value. This makes the outer function \(f(u) = |u|\). Here, \(u\) is a placeholder for whatever input it receives—specifically, \(1-x^3\) in our case. By recognizing this, you can bridge your way towards constructing the entire composite function effectively.

Absolute Value Function

The Absolute Value Function is unique. It primarily focuses on the magnitude or distance of a number from zero on a number line, completely ignoring its sign. In simpler terms, \(|a|\) equals \(a\) if \(a\) is positive or zero, and \(-a\) if \(a\) is negative.

• It removes any negative sign from its input.
• It is non-negative, as the absolute value is always zero or positive.
• Symbolized by two vertical lines, like \(|x|\).

In our problem, it checks the result of \(1-x^3\) and ensures the value \(H(x)\) remains positive or zero.

Function Composition

Function Composition is an exquisite tool in mathematics enabling more complex functions through a layered approach. When you compose functions, you're basically plugging one function into another.

• This operation is noted as \(f \circ g\), read as "\(f\) composed with \(g\)".
• Order matters: \(f(g(x))\) isn't always the same as \(g(f(x))\).

For our exercise, the composition is \(f(g(x)) = |1 - x^3|\). You take \(g(x) = 1 - x^3\) and feed it into \(f(u) = |u|\). This produces \(|1 - x^3|\), perfectly aligning with our function \(H(x)\). Function composition simplifies complex operations, just like assembling puzzle pieces.