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Chapter 1: Problem 45

Express the function in the form \(f \circ g .\) $$ F(x)=\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}} $$

Short Answer

Expert verified
The function is expressed as \( f(g(x)) \) where \( f(u) = \frac{u}{1+u} \) and \( g(x) = \sqrt[3]{x} \).

Step by step solution

01

Identify the Inner and Outer Functions

First, we need to express the original function in terms of inner and outer functions. Let's set \[ g(x) = \sqrt[3]{x} \].This simplifies our observation to a more compact form using a common expression.
02

Form the Outer Function

With the inner function identified, we express the given function with respect to this inner function. Based on the expression \[ F(x) = \frac{g(x)}{1+g(x)} \], we can let \[ f(u) = \frac{u}{1+u} \], where \( u = g(x) \). Here, \( u = \sqrt[3]{x} \) acts as the intermediary step.
03

Combine Inner and Outer Functions

Express the original function \( F(x) \) as a composition of the two identified functions:\[ F(x) = f(g(x)) \].Here, \( g(x) = \sqrt[3]{x} \) is the inner function and \( f(u) = \frac{u}{1+u} \) is the outer function.

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Key Concepts

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

Inner Function
The concept of an "inner function" is integral to understanding function composition. When given a complex function to express as a combination of simpler functions, the inner function is one of the components you identify first.
In the exercise, the function \( g(x) = \sqrt[3]{x} \) is identified as the inner function. This means \( g(x) \) is part of the more complex equation, working internally.
Think of the inner function as the initial input-transformer within the composite setup:
  • It takes \( x \) as its input.
  • Performs an operation, here taking the cube root of \( x \).
  • Produces an output that is then passed to the outer function.
The inner function plays a crucial role because it often dictates the transformation structure of the entire composite function sequence.
Outer Function
Once the inner function \( g(x) \) is defined, the next step in function composition is determining the "outer function". This function operates on the output generated by the inner function.
In our exercise, \( f(u) = \frac{u}{1+u} \) is designated as the outer function. Here, \( u \) represents the result from the inner function \( g(x) = \sqrt[3]{x} \).
The outer function is like the primary operator in your composition:
  • It receives the output of the inner function as its input.
  • Further processes this input to generate the final output.
  • Involves operations that may combine or further manipulate the intermediate result.
The outer function completes the process by wrapping around the transformation initiated by the inner function. It is the last step that transforms the specified input into the final computed value.
Composite Function
"Composite function" is a pivotal idea in mathematics, allowing us to simplify and solve complex equations by layering simple functions.
This composition involves combining an inner function and an outer function, where the composite function effectively chains the individual transformations.
For our exercise, the composite function is expressed as \( F(x) = f(g(x)) \):
  • \( g(x) \) deals with the initial input transformation.
  • \( f(g(x)) \) nests the transformation within a further operation, performed by \( f(u) \).
  • The full expression results in the fraction \( \frac{\sqrt[3]{x}}{1+\sqrt[3]{x}} \).
Composite functions facilitate review and breakdown of intricate operations into manageable parts. They are foundational in calculus and higher-level math, offering a roadmap for tackling layered problem-solving challenges.

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Most popular questions from this chapter

Find the exact value of each expression. $$ \begin{array}{llll}{\text { (a) } \sin ^{-1}(-1 / \sqrt{2})} & {\text { (b) } \cos ^{-1}(\sqrt{3} / 2)} & {} & {}\end{array} $$

Find the exact value of each expression. $$ \text { (a) } \arcsin (\sin (5 \pi / 4)) \quad \text { (b) } \cos \left(2 \sin ^{-1}\left(\frac{5}{13}\right)\right) $$

Solve each inequality for \(x\) $$ \text { (a) } 1

Find the exact value of each expression. $$ \begin{array}{lll}{\text { (a) } \tan ^{-1} \sqrt{3}} & {\text { (b) } \arctan (-1)} & {}\end{array} $$

Let \(f\) and \(g\) be linear functions with equations \(f(x)=m_{1} x+b_{1}\) and \(g(x)=m_{2} x+b_{2} .\) Is \(f \circ g\) also a linear function? If so, what is the slope of its graph?
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