Problem 49 - Algebra and Trigonometry

Question

\(45-50\) Express the function in the form \(f \circ g\) $$ H(x)=\left|1-x^{3}\right| $$

Step-by-Step Solution

Answer

The function is expressed as \(f \circ g\) where \(f(x) = |x|\) and \(g(x) = 1-x^3\).

Step 1: Identify the Outside and Inside Functions

To express function \(H(x) = \left|1-x^3\right|\) in the form \(f \circ g\), we should identify two functions such that \(f(x)\) is applied to \(g(x)\). Here, the absolute value operation is applied to the expression inside, suggesting that we can let \(f(x) = |x|\) and \(g(x) = 1-x^3\).

Step 2: Define Function \(g\)

We found that \(g(x) = 1 - x^3\). This function performs the operation on \(x\) before applying the absolute value.

Step 3: Define Function \(f\)

Function \(f\) is defined as \(f(x) = |x|\). It takes the result from \(g(x)\) and applies the absolute value operation to it.

Step 4: Combine Functions in Composition

The function composition \(f \circ g\) implies \(f(g(x))\). Thus, substituting our definitions, we have \(f(g(x)) = |g(x)| = |1-x^3|\), which matches the original \(H(x)\).

Key Concepts

Absolute Value Function

An absolute value function is a special mathematical function denoted by the symbol \(|x|\). It is used to determine the magnitude or distance of a number from zero on the number line, disregarding its sign.

When you observe \(|3|\), it means the distance of 3 from zero, which is simply 3. Similarly, \(|-3|\) is also 3 because distance can't be negative.

To break it down more:

• If \(x\) is positive, then \(|x| = x\)
• If \(x\) is negative, then \(|x| = -x\)

In the context of the given exercise, \(f(x) = |x|\). This function applies the absolute value to whatever result it receives from the inner function \(g(x)\). It ensures that the composite function \(H(x)\) always yields a non-negative outcome.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Typically, it looks something like this: \(c_n x^n + c_{n-1} x^{n-1} + \, ... \, + c_1 x + c_0\).

In simple forms:

• Polynomials can have terms like \(x^3\), \(x^2\), \(x\), and constants.
• Each term has a coefficient, which is basically the number in front of the variable or term.

For our exercise, the inner function \(g(x) = 1-x^3\) is a specific type of polynomial function. It includes a cube term \(-x^3\) and a constant term \(1\). This function is crucial as it defines the transformation that happens to \(x\) before the absolute value is applied in the composition.

Composition of Functions

Function composition is a process where one function is applied to the results of another function. This is represented as \(f \circ g\), which means \(f(g(x))\).

It works like this:

• First, you apply the function \(g\) to \(x\).
• Then, you take the output from \(g\) and use it as the input for function \(f\).

In the solved exercise, we composed two functions: \(f(x) = |x|\) and \(g(x) = 1-x^3\). This composition \(f(g(x))\) becomes \(|1-x^3|\), which was our original function \(H(x)\).

This step-by-step transformation shows how two seemingly separate mathematical operations combine into a single process to yield the same original function values.