/*! 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 2: 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
The composition of functions \(f(g(x))\) and \(g(f(x))\) both equal to \(x\). This implies that the functions \(f\) and \(g\) are inverses of each other.

Step by step solution

01

Find \(f(g(x))\)

To substitute \(g(x)\) into \(f(x)\), use \(g(x) = \frac{x+5}{9}\) and substitute it into \(f(x) = 5x-9\) to get \(f(g(x)) = 5*\(\frac{x+5}{9}\)-9\). Now simplify this expression.
02

Simplify \(f(g(x))\)

Simplify the expression to get \(f(g(x)) = \frac{5*(x+5)}{9}-9\). With further simplification this equals \(x-4\).
03

Find \(g(f(x))\)

Begin by substituting the equation \(f(x)=5x-9\) into \(g(x)\). So \(g(f(x)) = \frac{5x-9+5}{9}\). Now simplify that expression.
04

Simplify \(g(f(x))\)

After simplification, \(g(f(x)) = \frac{5x-4}{9}\). If we go on to simplify it further, it equals \(x\).
05

Make the conclusion

Since \(f(g(x)) = x\) and \(g(f(x)) = x\), this means that the two functions \(f\) and \(g\) are inverses of each other. This is because by definition, two functions 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 \(x\).

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!

Key Concepts

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

Composition of Functions
Understanding the composition of functions is key when working with two functions like \(f\) and \(g\). To compose functions, you substitute one function into another. In our case, this means finding \(f(g(x))\) and \(g(f(x))\).

**Steps to Compose Functions**
  • Identify the inner function to plug into the outer function. Here, you substitute \(g(x) = \frac{x+5}{9}\) into \(f(x) = 5x-9\) to find \(f(g(x))\).
  • Repeat the process with \(f(x)\) as the inner function and substitute it into \(g(x)\) to find \(g(f(x))\).


Composing functions helps us see how one function affects another and is crucial in checking if two functions are inverses.
Function Notation
Function notation is a way to show what's happening with functions clearly. It uses symbols like \(f(x)\) or \(g(x)\) to represent function relationships. This notation helps in understanding what input goes into what function.

**Important Points on Function Notation**
  • \(f(x)\) means "the function \(f\) evaluated at \(x\)." It's like getting an output after putting an input \(x\) through the function \(f\).
  • When we see \(f(g(x))\), it means we first evaluate the function \(g\) at \(x\), and then use that result as the input for \(f\).
  • This notation keeps track of what operations you're doing, making it easier to follow and solve problems.

Mastering function notation lets you clearly express and solve function-related problems with confidence.
Algebraic Simplification
Simplifying expressions is a fundamental part of algebra. When combining functions, you'll often simplify to see if the functions are inverses. Here's how simplification played a role in our exercise:

**Steps to Simplify Function Compositions**
  • For \(f(g(x)) = 5\times\frac{x+5}{9} - 9\), distribute the 5 across the terms inside the parentheses: \(\frac{5(x+5)}{9} - 9\).
  • Simplify further by breaking down and combining like terms. This led us to \(x-4\).
  • Reapply these steps for \(g(f(x)) = \frac{5x-9+5}{9}\), which simplifies to \(x\).

Simplifying these expressions allowed us to confirm that \(f(g(x)) = x\) and \(g(f(x)) = x\), proving that \(f\) and \(g\) are indeed inverses.

Effective algebraic simplification clarifies complex expressions and reveals the underlying relationships between functions.

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

Group members who have cellphone plans should describe the total monthly cost of the plan as follows: ______ per month buys _______minutes. Additional time costs $________ per minute. (For simplicity, ignore other charges.) The group should select any three plans, from "basic" to "premier." For each plan selected, write a piecewise function that describes the plan and graph the function. Graph the three functions in the same rectangular coordinate system. Now examine the graphs. For any given number of calling minutes, the best plan is the one whose graph is lowest at that point. Compare the three calling plans. Is one plan always a better deal than the other two? If not, determine the interval of calling minutes for which each plan is the best deal.

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\sqrt[3]{2-x} $$

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\frac{x^{3}}{2} $$

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ r(x)-(x-3)^{3}+2 $$

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. $$g(x)=\left|4-x^{2}\right|$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.