/*! 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 29 Find a formula for the inverse f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=6+x^{3} $$

Short Answer

Expert verified
The inverse function of the given function \(f(x) = 6 + x^3\) is \(f^{-1}(x) = \sqrt[3]{x - 6}\).

Step by step solution

01

Replace function notation with y

Replace \(f(x)\) with \(y\) to make the equation easier to manipulate: $$ y = 6 + x^3 $$
02

Swap the roles of x and y

Swap the x and y variables in the equation: $$ x = 6 + y^3 $$
03

Solve the equation for y

We want to get y by itself on one side of the equation, so first subtract 6 from both sides: $$ x - 6 = y^3 $$ Now take the cube root of both sides to isolate y: $$ y = \sqrt[3]{x - 6} $$
04

Write the resulting equation using the inverse function notation

Replace y with the inverse function notation \(f^{-1}(x)\): $$ f^{-1}(x) = \sqrt[3]{x - 6} $$ The inverse function is given by: $$ f^{-1}(x) = \sqrt[3]{x - 6} $$

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

Find all numbers \(x\) that satisfy the given equation. $$ (\log (3 x)) \log x=4 $$

Explain why $$ 1+\log x=\log (10 x) $$ for every positive number \(x\)

Explain why $$ (1+\log x)^{2}=\log \left(10 x^{2}\right)+(\log x)^{2} $$ for every positive number \(x\)

Suppose \(f\) is a function with exponential decay. Explain why the function \(g\) defined by \(g(x)=\frac{1}{f(x)}\) is a function with exponential growth.

The most intense recorded earthquake in the state of New York occurred in 1944 ; it had Richter magnitude \(5.8 .\) The most intense recorded earthquake in Minnesota occurred in \(1975 ;\) it had Richter magnitude \(5.0 .\) Approximately how many times more intense was the 1944 earthquake in New York than the 1975 earthquake in Minnesota?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.