/*! 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 72 Find the inverse of each functio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 72

Find the inverse of each function. $$f(x)=2 \sqrt[3]{x}$$

Short Answer

Expert verified
f^{-1}(x) = \left( \frac{x}{2} \right)^3

Step by step solution

01

Rewrite the Function as y

Begin by rewriting the function using y instead of f(x). This gives us the equation: \[ y = 2 \sqrt[3]{x} \]
02

Solve for x

Next, we need to solve this equation for x. Start by isolating the cube root term. Divide both sides by 2: \[ \frac{y}{2} = \sqrt[3]{x} \]
03

Cube Both Sides

To eliminate the cube root, cube both sides of the equation: \[ \left( \frac{y}{2} \right)^3 = x \]
04

Rewrite Inverse Function

Now that we have solved for x, rewrite the equation with x in terms of y as the inverse function f^{-1}(x): \[ f^{-1}(x) = \left( \frac{x}{2} \right)^3 \]

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.

Function Notation
Function notation is a way to represent functions in mathematics. When you see something like \( f(x) \), it means that f is a function of x. This helps us understand what the function does and how we use it.
For instance, if you see \( f(x) = 2 \sqrt[3]{x} \), this tells us that the function f takes an input x, applies the cube root to it, and then multiplies the result by 2.
Remember, function notation helps us clearly define and work with functions, which is crucial in solving problems like finding inverse functions.
Solving Equations
Solving equations is about finding the value of your variable that makes the equation true. It involves a step-by-step approach to isolate the variable.
For example, suppose we start with \( y = 2 \sqrt[3]{x} \). To solve for x, we need to isolate it by performing operations in reverse order.
First, divide both sides by 2: \( \frac{y}{2} = \sqrt[3]{x} \).
Next, to get rid of the cube root, you cube both sides: \( \left( \frac{y}{2} \right)^3 = x \).
This gives you the solution for x, which is essential for finding the inverse function.
Cube Roots
A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).
In the context of functions, dealing with cube roots often comes up. For example, in\( f(x)= 2 \sqrt[3]{x} \), we see that the function involves taking the cube root of x.
When finding the inverse, we must undo this operation. This is why we cubed both sides in the solution: to cancel out the cube root and isolate x. Understanding cube roots is crucial when dealing with various types of functions, especially when finding inverses or solving equations involving roots.

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

Let \(f(x)=x^{2}\) and \(g(x)=x+5 .\) Determine whether each of these statements is true or false. $$(f+g)(4)=21$$

Classify each function as either a linear, constant, quadratic, square-root, or absolute value function. $$f(x)=-5 x^{2}+8 x+2$$

Discussion. Let \(f(x)=\sqrt{x-4}\) and \(g(x)=\sqrt{x-8} .\) Find the domains of \(f, g,\) and \(f+g\).

Solve each problem. Sail area-displacement ratio. To find the sail areadisplacement ratio \(S,\) first find \(y,\) where \(y=(d / 64)^{2 / 3}\) and \(d\) is the displacement in pounds. Next find \(S,\) where \(S=A / y\) and \(A\) is the sail area in square feet. a) For the Pacific Seacraft \(40, A=846\) square feet \(\left(\mathrm{ft}^{2}\right)\) and \(d=24,665\) pounds. Find \(S\) b) For a boat with a sail area of \(900 \mathrm{ft}^{2},\) write \(S\) as a function of \(d\) c) For a fixed sail area, does \(S\) increase or decrease as the displacement increases?

Find the inverse of each function. $$f(x)=\sqrt[3]{2 x-1}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.