Problem 36 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=(5 x-1)^{3} $$

Step-by-Step Solution

Answer

Functions are \( f(x) = x^3 \) and \( g(x) = 5x - 1 \).

Step 1: Identifying Inner Function

To express the function \( h(x) = (5x - 1)^3 \) in the form \( h(x) = f(g(x)) \), first identify the inner function \( g(x) \). Notice that \( 5x - 1 \) is inside the exponent, suggesting it can be our inner function, \( g(x) = 5x - 1 \).

Step 2: Identifying Outer Function

After choosing \( g(x) = 5x - 1 \), identify \( f(x) \) by expressing \( h(x) \) in terms of \( g(x) \). We have \( h(x) = (5x - 1)^3 \), which can be rewritten using \( g(x) \): \( h(x) = g(x)^3 \). Therefore, the outer function is \( f(x) = x^3 \).

Step 3: Verify the Composition

Now verify the composition by substituting back into \( f(g(x)) \). Substitute \( g(x) = 5x - 1 \) into \( f(x) = x^3 \) to get \( f(g(x)) = (5x - 1)^3 \), which matches the original function \( h(x) \). This confirms that the functions \( f(x) = x^3 \) and \( g(x) = 5x - 1 \) correctly express \( h(x) \) as a composition.

Key Concepts

Inner Function

Understanding the concept of an inner function is essential when dealing with function composition. Think of it as the core of a composite function. It is the function that you substitute into another function.
In our given problem, the function is expressed as \( h(x) = (5x - 1)^3 \). We aim to express this as \( h(x) = f(g(x)) \). To identify the inner function \( g(x) \), observe what is inside the "main" operation—in this case, the expression inside the cube.
Thus, the inner function here is \( g(x) = 5x - 1 \).
By recognizing \( g(x) \), you're setting up the stage for the outer function and the entire composition.

• Think of the inner function as a stepping stone, giving structure to the function you're analyzing.
• In practical terms, the inner function is often found where variables are transformed or manipulated before undergoing further operations.

Outer Function

The outer function in a composition acts on the result of the inner function. It is the main operation or transformation that is applied to the output from the inner function.
In our example, once we establish that the inner function is \( g(x) = 5x - 1 \), the task is to determine how this output is used in \( h(x) = f(g(x)) \).
Looking at \( h(x) = (5x - 1)^3 \), notice that the operation after obtaining \( 5x - 1 \) is cubing the result.
Hence, the outer function is \( f(x) = x^3 \).

• The outer function modifies the output of the inner function, providing the final form of the composition.
• Identifying the outer function involves recognizing how the expression given by the inner function is further manipulated.

This is crucial to completing the expression in its composite form, \( h(x) = f(g(x)) \).

Algebraic Functions

Algebraic functions are building blocks in mathematics, involving operations such as addition, subtraction, multiplication, division, and roots applied to variables.
They are important because they compose more complex functions like polynomials, rational, and root functions.
In the context of function composition, as seen in our example with \( h(x) = (5x - 1)^3 \), algebraic functions allow us to break down and rearrange expressions into manageable parts.

• These functions help simplify expressions and solve equations by expressing complex relationships in terms of basic operations.
• Algebraic manipulation involves rearranging terms, factoring, expanding, and other operations that can transform the function’s form.

Recognizing and understanding different algebraic forms is key to mastering topics like function composition, as it grants the ability to deconstruct and reconstruct functions effectively.