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Chapter 3: Problem 5

Each of the following functions may be viewed as a composite function \(h(x)=f(g(x))\). Find \(f(x)\) and \(g(x)\). $$h(x)=\left(x^{3}+8 x-2\right)^{5}$$

Short Answer

Expert verified
Here, let \( f(x) = x^5 \)and \( g(x) = x^3 + 8x - 2 \).

Step by step solution

01

- Identify the Outer Function

The given function is \[h(x)=(x^3 + 8x - 2)^5.\] Notice that this can be seen as a composite function where the outer function is a power function. So, let鈥檚 express this as \[h(x) = [u(x)]^5.\]
02

- Define the Inner Function

Now observe the inner part of the composite function. We can define the inner function \(g(x)\) as the expression inside the power function. So, \[g(x) = x^3 + 8x - 2.\]
03

- Define the Outer Function

The outer function \(f(x)\) is the function that raises its input to the 5th power. Therefore, \[f(x) = x^5.\]
04

- Verify the Composite Function

To verify, we need to ensure that \(h(x) = f(g(x))\). If \(g(x) = x^3 + 8x - 2\) and \(f(x) = x^5\), then \[f(g(x)) = f(x^3 + 8x - 2) = (x^3 + 8x - 2)^5,\] which matches the original function \(h(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Outer Function
A composite function is made up of an **outer function** and an **inner function**. In the exercise, we have the function h(x) = (x^3 + 8x - 2)^5.The outer function is the one that would act last if the function were to be decomposed into a sequence of operations. In this case, the outer function is raising the expression inside to the power of 5. So, we define the outer function as f(x) = x^5.This means that if we know the value inside (which the inner function provides), we can compute the final result by simply evaluating f(x).
Inner Function
The **inner function** is the part of the composite function that gets input first. In our example, it is the equation found within the brackets of our outer function. For the functionh(x) = (x^3 + 8x - 2)^5,we need to identify the expression inside the power, which isx^3 + 8x - 2.Thus, we denote our inner function asg(x) = x^3 + 8x - 2.The inner function is responsible for preparing the input for the outer function, which then applies its operation to this input.
Function Composition
Putting everything together, we get a composite function, where one function is applied to the result of another. This process is known as **function composition**. Given our example,h(x) = (x^3 + 8x - 2)^5,we can write this as a composition of the outer function and the inner function, specificallyh(x) = f(g(x)).Here, f(x) = x^5 and g(x) = x^3 + 8x - 2. The composition means that first we use the inner function: g(x), then, we apply the outer function: f(x). So, f(g(x)) becomes f(x^3 + 8x - 2), which simplifies to(x^3 + 8x - 2)^5.By understanding function composition, we can easily break down and analyze complex functions into simpler parts.

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Most popular questions from this chapter

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