/*! 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 7 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 7

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

Short Answer

Expert verified
After performing the compositions of \(f(x)\) and \(g(x)\), i.e., computing \(f(g(x))\) and \(g(f(x))\), simple the expressions. If both results are \(x\), then the functions are inverses of each other. If not, they are not inverses.

Step by step solution

01

Find the Composition \(f(g(x))\)

Substitute the function \(g(x)\) into \(f(x)\) to find \(f(g(x))\): \(f(g(x))=f(\frac{3}{x}+4)=\frac{3} {\left(\frac{3}{x}+4\right) - 4}\). Simplify this to obtain the function \(f(g(x))\).
02

Find the Composition \(g(f(x))\)

Substitute the function \(f(x)\) into \(g(x)\) to find \(g(f(x))\): \(g(f(x))=g(\frac{3}{x - 4})=\frac{3} {\left(\frac{3}{x - 4}\right)} + 4\). Simplify this to get the function \(g(f(x))\).
03

Determine if the functions are inverses

The functions \(f\) and \(g\) are inverses of each other if and only if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domain of the functions. Compare the compositions of the functions to \(x\).

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鈥檚 domain and range. $$(x+2)^{2}+y^{2}=16$$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} (x-2)^{2}+(y+3)^{2} &=4 \\ y &=x-3 \end{aligned}$$

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

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+8 x+4 y+16=0$$

Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.