Problem 75 - Precalculus Essentials

Question

Express the given function \(h\) as a composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=(3 x-1)^{4}$$

Step-by-Step Solution

Answer

The function \(h(x)\) can be expressed as the composition of \(f(x) = x^{4}\) and \(g(x) = 3x - 1\).

Step 1: Identify the Inner Function

The square operation in \(h(x)=(3x-1)^{4}\) is performed on the result of an operation on \(x\), which was to multiply \(x\) by 3 and subtract 1. Therefore, \(g(x) = 3x - 1\)

Step 2: Identify the Outer Function

After applying \(g\), the result is then raised to the fourth power. This operation, raising something to the fourth power, can be our function \(f\). So, \(f(x) = x^{4}\)

Step 3: Verify the Composition

To ensure that we got the correct decomposition, we perform the composition: \((f \circ g) (x) = f(g(x)) = f(3x-1) = (3x - 1)^{4}\) which is exactly the function \(h(x)\) we want to decompose

Key Concepts

Inner Function

When dealing with function composition, it's essential to identify components known as the **inner function** and the **outer function**. The inner function is the function that is applied first in the process of evaluating expressions. It determines the initial transformation of the input value in function composition.In the example given, we have the function:\[ h(x) = (3x - 1)^4 \]Here, before dealing with the exponent, we first perform the operation within the brackets. This internal manipulation is defined by the inner function, which is:- **Inner Function**: \( g(x) = 3x - 1 \)This means that for any input \( x \), you first evaluate \( g(x) \) to transform \( x \) into \( 3x - 1 \). After this initial step, the transformation through the inner function, the next step is applying the outer function, which will complete the composition.

Outer Function

Once the inner function has done its job, the next phase in function composition is managed by the **outer function**. The outer function is responsible for taking the result of the inner function and performing additional operations to get the final output.In our example:\[ h(x) = (3x - 1)^4 \]The transformation of \( 3x - 1 \) is carried out by the outer function. This function takes what the inner function produces and applies further transformation, which is in this case raising the result to the 4th power.- **Outer Function**: \( f(x) = x^4 \)So, after calculating \( g(x) = 3x - 1 \), the result is then processed by \( f \) by raising it to the 4th power. This sequential processing of inputs leads to the complete function output, which aligns perfectly with our original function \( h(x) \).

Decomposition of Functions

Decomposition of functions is when we break down a complex function into simpler parts, usually called the inner and outer functions. This approach is not only useful in understanding the structure of the function but also in simplifying calculations and problem-solving.For the example with \( h(x) = (3x - 1)^4 \), we saw that:- The **inner function** is \( g(x) = 3x - 1 \)- The **outer function** is \( f(x) = x^4 \)The process of decomposition involves listing these components such that applying the inner function first (\( g(x) \)) followed by the outer function (\( f(x) \)) results in the original function \( h(x) \). Function decomposition helps to unravel complex expressions and enables a more straightforward evaluation and manipulation of mathematical expressions. When learning the decomposition of functions, always remember:

• Identify what operation happens first to find the inner function.
• See what operation is applied afterward, which defines the outer function.