Problem 87 - College Algebra with Modeling and Visualization

Question

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=\left(x^{3}-1\right)^{2} $$

Step-by-Step Solution

Answer

Let \(f(x) = x^3 - 1\) and \(g(x) = x^2\).

Step 1: Understand the Composition Context

The notation \((g \circ f)(x)\) means that the function \(f(x)\) is applied first, and then the function \(g(x)\) is applied to the result. So, we need to express \(h(x) = (x^{3}-1)^{2}\) in this composed form.

Step 2: Determine the Inner Function \(f(x)\)

Let's consider \(f(x) = x^3 - 1\). The goal here is to simplify the expression inside the parenthesis of \(h(x)\). This choice simplifies the expression to \(f(x)\).

Step 3: Define the Outer Function \(g(x)\)

Now, recognizing that \((x^{3}-1)^{2}\) can be expressed as \(g(f(x))\), we define \(g(x) = x^2\). This means that once we apply \(f(x)\), we then apply \(g(x)\) to the result, which squares it.

Step 4: Verify the Composition

Compose the functions to ensure correctness: If \(f(x) = x^3 -1\) and \(g(x) = x^2\), then \(h(x) = g(f(x)) = g(x^3 - 1) = (x^3 - 1)^2\), which matches the original \(h(x)\).

Key Concepts

Composition of Functions

When we talk about the composition of functions, we're dealing with a powerful mathematical process that allows us to combine two or more functions to create a new function. This new function represents the outcome of applying one function to the results of another.
For instance, if we have two functions, say function \( f(x) \) and function \( g(x) \), composing them involves applying \( f(x) \) first, and then applying \( g(x) \) to the result of \( f(x) \). This gives us what is called the composite function, notated as \((g \circ f)(x)\).

Understanding function composition can be extremely helpful in many mathematical and real-world applications. It allows us to simplify complex systems into manageable parts.

• Determine the sequence of function application
• Apply the functions step-by-step
• Simplify using the composition of simpler functions

Recognizing how functions interact and modify each other through composition is key to solving complex mathematical problems with ease.

Inner Function

The inner function is simply the first function we apply to our input value in a composition of functions. In the given exercise, we needed to find appropriate functions \( f(x) \) and \( g(x) \) such that their composition \((g \circ f)(x)\) gives us the function \( h(x) \).
In this example, by defining the inner function as \( f(x) = x^3 - 1 \), we are specifying that we first transform our input \( x \) by applying \( f(x) \). This choice is derived from the requirement to simplify the inside of the given composite function to match \((x^3 - 1)\).

The inner function sets the stage for its result to be further transformed by the outer function, making each step in the computation clear:

• Identify the part of the composite equation that is computed first
• Ensure that the output can be used as an input for the next step

Choosing the correct inner function is essential to ensure the overall function composition correctly matches \( h(x) \).

Outer Function

The outer function in function composition is the last function applied to the output of the inner function. In the exercise, once we chose \( f(x) = x^3 - 1 \) as our inner function, we then needed an outer function \( g(x) \) that would transform \( f(x) \) into \( h(x) = (x^3 - 1)^2 \).
By selecting the outer function as \( g(x) = x^2 \), we achieve this transformation. The role of the outer function is to take the output from the inner function and apply a new rule, which in this case squares the result.

This completes the function composition process:

• Accepts the output from the inner function as input
• Applies another transformation to yield the final output

With both functions working together, the composite function \( h(x) \) is accurately defined, demonstrating the power of structuring computations through function composition.