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Chapter 0: Problem 47

Find two functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\). (There are many correct answers.) $$h(x)=(2 x+1)^{2}$$

Short Answer

Expert verified
The function \(f(x)=x^2\) and \(g(x)=2x+1\) are two functions such that \((f \circ g)(x)=h(x)\). However, there are multiple correct answers to this exercise.

Step by step solution

01

Understanding the composite function

Composite function is an application of one function to the results of another. It's written as \(f \circ g(x)\) which is same as \(f(g(x))\). It means you first apply \(g(x)\) and then apply \(f(x)\) to the result of \(g(x)\). In our case, we want to decompose \(h(x)\) into two such functions.
02

Select a function g(x)

To simplify things, we first aim to get the inner part of the function \(h(x)\) i.e. \(2x+1\). Thus, we can choose \(g(x) = 2x+1\).
03

Select a function f(x)

After applying \(g(x)\) to \(x\), we got \(2x+1\). Looking at \(h(x)\) again, the \(2x+1\) is squared. Thus, for the second function, we can choose \(f(x) = x^2\).
04

Check if the selection of f(x) and g(x) is correct

We need to check if our selected functions give us \(h(x)\) on composition. Let's apply \(g(x)\) to \(x\), we get \(2x+1\). Now let's apply this result to \(f(x)\) i.e. \((2x+1)^2\) = \(h(x)\). Thus, our selected functions \(f(x)\) and \(g(x)\) are correct. Three things to note here: 1) \(h(x)\) can be decomposed into many other combinations of \(f(x)\) and \(g(x)\), 2) for different combinations of \(f(x)\) and \(g(x)\), the properties of these functions will vary, and 3) while \(f \circ g(x)\) = \(h(x)\) is true, the reverse that \(g \circ f(x)\) = \(h(x)\) is not necessarily true.

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Key Concepts

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

Composite Function
Composite functions are like layered processes where one function provides output to be used by another function. Imagine baking a cake. First, you mix the ingredients, which is similar to applying one function. Then, you bake the mixture, which is like applying a second function to the previously obtained results.

Symbolically, a composite function is expressed as \( f \circ g(x) \), which means you apply \( g(x) \) first and then \( f(x) \) to the result, effectively making it \( f(g(x)) \).

This technique allows for complex operations to be broken down into simpler tasks. Such a breakdown is vital in math because it helps us manage and solve intricate problems step-by-step.
Decomposition of Functions
Decomposing functions is the process of breaking down a complex function into simpler, more manageable parts. Think of it as dismantling a machine to understand how it works by inspecting each component. In mathematics, this enables us to better analyze and solve functions by focusing on simpler terms.

Take, for example, a function \( h(x) = (2x+1)^2 \). Decomposing it involves identifying two simpler functions:
  • \( g(x) = 2x+1 \), which represents the inner operation.
  • \( f(x) = x^2 \), which represents the outer operation or the squaring process.
This decomposition results in the composite function \( f \circ g(x) \), such that applying \( g(x) \) followed by \( f(x) \) will reconstruct the original function \( h(x) \). It's important to notice that there are multiple ways to decompose a given function based on different perspectives of the inner and outer operations.
Trigonometric Functions
Trigonometric functions are special functions that relate angles of a triangle to the ratios of its sides. Common trig functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).

These functions are mainly used in geometry, physics, and engineering. They are essential for understanding waves, circular motion, and oscillatory behavior.

While this exercise isn't directly related to trig functions, understanding composite operations is crucial when dealing with trigonometric functions. For example, if you have \( h(x) = \sin(x^2) \), you can view it as a composition of functions just like \( f(x) = \sin(x) \) and \( g(x) = x^2 \).

By recognizing how compositions can apply to trig functions, we can more easily tackle complex trigonometric equations by breaking them into simpler pieces.

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Most popular questions from this chapter

Path of a Ball The height \(y\) (in feet) of a baseball thrown by a child is $$ y=-\frac{1}{10} x^{2}+3 x+6 $$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

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In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{3 x+4}{5} $$

The total numbers \(f\) (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Federal Highway Administration) $$ \begin{array}{|c|c|} \hline 0 \text { Year, } t & \text { Miles traveled, } f(t) \\ \hline 5 & 2423 \\ 6 & 2486 \\ 7 & 2562 \\ 8 & 2632 \\ 9 & 2691 \\ 10 & 2747 \\ 11 & 2797 \\ 12 & 2856 \\ \hline \end{array} $$ (a) Does \(f^{-1}\) exist? (b) If \(f^{-1}\) exists, what does it mean in the context of the problem? (c) If \(f^{-1}\) exists, find \(f^{-1}\) (2632). (d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year was 2747 billion, would \(f^{-1}\) exist? Explain.

Digital Camera Sales The factory sales \(f\) (in millions of dollars) of digital cameras in the United States from 1998 through 2003 are shown in the table. The time (in years) is given by \(t\), with \(t=8\) corresponding to 1998 . (Source: Consumer Electronincs Association) $$ \begin{array}{|c|c|} \hline \text { Year, } t & \text { Sales, } f(t) \\ \hline 8 & 519 \\ 9 & 1209 \\ 10 & 1825 \\ 11 & 1972 \\ 12 & 2794 \\ 13 & 3421 \\ \hline \end{array} $$ (a) Does \(f^{-1}\) exist? (b) If \(f^{-1}\) exists, what does it represent in the context of the problem? (c) If \(f^{-1}\) exists, find \(f^{-1}(1825)\). (d) If the table was extended to 2004 and if the factory sales of digital cameras for that year was \(\$ 2794\) million, would \(f^{-1}\) exist? Explain.
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