/*! 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 5 Find \(f(g(x))\) and \(g(f(x))\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=5 x-9 \text { and } g(x)=\frac{x+5}{9}$$

Short Answer

Expert verified
Yes, the functions \(f(x) = 5x - 9\) and \(g(x) = \frac{x+5}{9}\) are inverses of each other as the compositions \(f(g(x))\) and \(g(f(x))\) both return \(x\).

Step by step solution

01

Calculate \(f(g(x))\)

Begin by plugging the function \(g(x) = \frac{x+5}{9}\) into the function \(f(x) = 5x - 9\). It would look like this: \(f(g(x)) = 5(\frac{x+5}{9}) - 9\). Simplify this to get \(f(g(x)) = x\).
02

Calculate \(g(f(x))\)

Now, you need to plug the function \(f(x) = 5x - 9\) into the function \(g(x) = \frac{x+5}{9}\). It would look like this: \(g(f(x)) = \frac{(5x - 9) + 5}{9}\). Simplify this to get \(g(f(x)) = x\).
03

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

Since the results from Step 1 and Step 2 are both \(x\), it can be concluded that \(f\) and \(g\) are indeed inverses of each other because \(f(g(x)) = x\) and \(g(f(x)) = x\). This implies that for every \(x\) in the domain, the compositions of \(f\)and \(g\) still return \(x\) as the result.

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

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+12 x-6 y-4=0$$

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=7$$

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$h(x)=|x-2|+|x+2|$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.