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Chapter 2: Problem 65

Express the function in the form \(f \circ g\). $$G(x)=\frac{x^{2}}{x^{2}+4}$$

Short Answer

Expert verified
\( G(x) = f(g(x)) \) with \( f(x) = \frac{x}{x+4} \) and \( g(x) = x^2 \).

Step by step solution

01

Identify inner function

The inner function, denoted as \( g(x) \), often simplifies the argument or the expression within the main function. Looking at \( G(x) = \frac{x^2}{x^2 + 4} \), the expression \( x^2 \) appears twice, hence we can isolate \( g(x) \) to be \( x^2 \).
02

Identify the outer function

Given \( g(x) = x^2 \), we need the outer function \( f(x) \) such that \( f(g(x)) = \frac{x^2}{x^2 + 4} \). Substitute \( g(x) \) to transform the expression: \( f(g(x)) = \frac{g(x)}{g(x) + 4} \). This implies \( f(x) = \frac{x}{x + 4} \).
03

Express the function in composite form

Now that we have \( g(x) = x^2 \) and \( f(x) = \frac{x}{x+4} \), we can express \( G(x) \) using the composite function as: \( G(x) = f(g(x)) = f(x^2) = \frac{x^2}{x^2 + 4} \). Thus, \( G(x) = f(g(x)) \) where \( f(x) = \frac{x}{x+4} \) and \( g(x) = x^2 \).

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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 the inner function in a composite function involves identifying the part of the function that can be thought of as the initial transformation or step.
The inner function is sometimes called "the input requirement" of the composite function as it prepares the 'input' for the next stage.
In our example, the function \( G(x) = \frac{x^2}{x^2 + 4} \), simplifies to get the inner function as \( g(x) = x^2 \).
  • Notice how \( x^2 \) is an expression that appears both in the numerator and the denominator of \( G(x) \).
  • This repeated use suggests that it acts as a building block.
This means the input prepared for further computation is \( g(x) \). Remember: the inner function is the part of the function's structure that you can repeatedly see being calculated first.
Outer Function
The outer function is the key to understanding how the final picture of composition is painted.
Once you have pinpointed and executed the inner function, you use it as the new variable or value that your main function will manipulate.
For our given \( G(x) = \frac{x^2}{x^2 + 4} \), after identifying \( g(x) = x^2 \), the outer function can be determined.
How? By thinking of \( f(g(x)) \) as \( \frac{g(x)}{g(x) + 4} \).
  • The resulting transformation helps us define the outer function \( f(x) = \frac{x}{x + 4} \).
  • This function operates on the solution of the inner task \( g(x) \).
In a bigger picture, the outer function shows how the output from the inner function gets utilized.
Function Composition
Function composition is a process of applying one function to the results of another. Imagine it as a series of steps where each stage builds on the output of the previous.
This sequential process is mathematically represented as \( f \circ g \), meaning \( f(g(x)) \).
For the function \( G(x) = \frac{x^2}{x^2 + 4} \), it was efficiently expressed as a composition by:
  • Identifying the inner function \( g(x) = x^2 \).
  • Defining the outer function \( f(x) = \frac{x}{x + 4} \).
  • Understanding that \( G(x) = f(g(x)) \) showcases how each function plays a role.
With this approach, we smoothly combine two separate functions to achieve the desired result.
Conceptually, it's like nesting operations, where the output of \( g(x) \) becomes the input for \( f(x) \).

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Most popular questions from this chapter

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(g(x)=x^{2}-x-20\) (a) \([-2,2]\) by \([-5,5]\) (b) \([-10,10]\) by \([-10,10]\) (c) \([-7,7]\) by \([-25,20]\) (d) \([-10,10]\) by \([-100,100]\)

Graphing Functions Sketch a graph of the function by first making a table of values. $$f(x)=\frac{x}{|x|}$$

A linear function is given. (a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$f(x)=\frac{1}{2} x+3$$

An appliance dealer advertises a \(10 \%\) discount on all his washing machines. In addition, the manufacturer offers a \(\$ 100\) rebate on the purchase of a washing machine. Let \(x\) represent the sticker price of the washing machine. (a) Suppose only the \(10 \%\) discount applies. Find a function \(f\) that models the purchase price of the washer as a function of the sticker price \(x\). (b) Suppose only the \(\$ 100\) rebate applies. Find a function \(g\) that models the purchase price of the washer as a function of the sticker price \(x\). (c) Find \(f \circ g\) and \(g \circ f .\) What do these functions represent? Which is the better deal?

An object is dropped from a high cliff, and the distance (in feet) it has fallen after \(t\) seconds is given by the function \(d(t)=16 t^{2}\) Complete the table to find the average speed during the given time intervals. Use the table to determine what value the average speed approaches as the time intervals get smaller and smaller. Is it reasonable to say that this value is the speed of the object at the instant \(t=3 ?\) Explain. $$\begin{array}{|c|c|c|} \hline t=a & t=b & \text { Average speed }=\frac{d(b)-d(a)}{b-a} \\ \hline 3 & 3.5 & \\ 3 & 3.1 & \\ 3 & 3.01 & \\ 3 & 3.001 & \\ 3 & 3.0001 & \\ \hline \end{array}$$
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