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Chapter 6: Problem 47

Find functions \(f\) and \(g\) so that \(f \circ g=H\).\(H(x)=(2 x+3)^{4}\)

Short Answer

Expert verified
Let \(g(x) = 2x + 3\) and \(f(u) = u^4\).

Step by step solution

01

Understanding the Composition

The composition of functions \(f \circ g\) means \(f(g(x))\). Therefore, we need to identify two functions \(f\) and \(g\) such that \(f(g(x))=H(x)=(2x+3)^4\).
02

Identify the Inner Function

Identify \(g(x)\) such that it forms part of the composite function. Notice that \(2x + 3\) is inside the power of 4. Let \(g(x) = 2x + 3\).
03

Identify the Outer Function

Given that \(g(x) = 2x + 3\), we need to determine \(f(u)\) where \(u = g(x)\). Since \(H(x) = (2x + 3)^4\), let \(f(u) = u^4\). Thus, \(f(u) = (2x + 3)^4\).
04

Verify the Functions

Verify the functions by composing \(f\) and \(g\). Substitute \(g(x) = 2x + 3\) into \(f(u) = u^4\). This yields \(f(g(x)) = (2x + 3)^4 \), confirming that \(f(g(x)) = H(x)\).

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Key Concepts

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

Composite Functions
Composite functions involve combining two functions such that the output of one function becomes the input of another. If you have functions named as \( f \) and \( g \), the composite function \( f \circ g \) means \( f(g(x)) \).
For example, if you have \( H(x) = (2x + 3)^4 \), you can create this composite form by breaking it into two simpler functions where one is nested inside the other.
Inner Function
The inner function is the function you apply first in a composite function. It can be thought of as the core or the starting point of the whole composition.
In our example problem, we observe the term \(2x + 3\) inside the power of 4. This inner function can be denoted as \( g(x) = 2x + 3 \).
Start with identifying these parts when dealing with composite functions as it guides you in structuring the outer function.
Outer Function
The outer function is applied after the inner function in composite functions. Essentially, the result of the inner function is plugged into this function.
Given our inner function \( g(x) = 2x + 3 \), note that the entire expression \( (2x + 3) \) is raised to the power of 4. Here, the outer function can be represented as \( f(u) = u^4 \).
By differentiating between these inner and outer functions, you can reassemble them to form the original composite function \( H(x) = (2x + 3)^4 \).
Function Verification
Function verification ensures that the proposed functions for \( f \circ g \) indeed produce the given function when composed.
To verify, substitute \( g(x) = 2x + 3 \) into \( f(u) = u^4 \).
That is, replace \( u \) in \( f(u) \) with \( g(x) \), resulting in \( f(g(x)) = (2x + 3)^4 \).
Doing this, if the original function is regained, then the chosen inner and outer functions are verified as correct.

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