/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Find \(f(g(x))\) and \(g(f(x))\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\sqrt[3]{x-4} \text { and } g(x)=x^{3}+4$$

Short Answer

Expert verified
The composite functions \(f(g(x))\) and \(g(f(x))\) both evaluate to \(x\). Therefore, the functions \(f(x)=\sqrt[3]{x-4}\) and \(g(x)=x^{3}+4\) are inverses of each other.

Step by step solution

01

Compute \(f(g(x))\)

To find \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). That is, wherever there is \(x\) in \(f(x)\), replace it with \(g(x)\). So, \(f(g(x)) = f(x^{3}+4) = \sqrt[3]{x^{3}+4 - 4} = \sqrt[3]{x^{3}} = x\)
02

Compute \(g(f(x))\)

Similarly, to find \(g(f(x))\), insert \(f(x)\) into \(g(x)\). So, \(g(f(x)) = g(\sqrt[3]{x - 4}) = (\sqrt[3]{x - 4})^{3} + 4 = x - 4 + 4 = x\)
03

Check if \(f\) and \(g\) are inverses

Having found that both \(f(g(x)) = x\) and \(g(f(x)) = x\), it can be concluded that the functions \(f\) and \(g\) are indeed inverses of each other correct to the given exercise context. Inverses undo each other's operations - in this case, cubing and the cube-root. Hence, the conclusion.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+3)^{2}+(y-2)^{2}=4 $$

In Exercises \(105-108,\) you will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used, \(x .\) Then find and interpret \(S(30)\)

Exercises \(129-131\) will help you prepare for the material covered in the next section. The function \(C(t)=20+0.40(t-60)\) describes the monthly cost, \(C(t),\) in dollars, for a cellphone plan for \(t\) calling minutes, where \(t>60 .\) Find and interpret \(C(100)\)

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even, odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h\) definitely an odd function?

The function $$ f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95 $$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a \([0,100,5]\) by \([0,40,2]\) viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the \([\mathrm{TABLE}]\) or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.