/*! 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 54 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 54

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). $$f(x)=\frac{x^{3}}{2}$$

Short Answer

Expert verified
Yes, the function \( f(x)=\frac{x^{3}}{2} \) is one-to-one and hence has an inverse that is also a function.

Step by step solution

01

Graph the function

Using a graphing utility, plot the function \( f(x)=\frac{x^{3}}{2} \). You will find that it creates a curving line, starting at negative infinity, passing through the origin, and proceeding towards positive infinity. The graph is symmetric with respect to the origin.
02

Conduct the horizontal line test

Draw horizontal lines across the graph. If each horizontal line intersects the graph at only one point, that verifies the existence of an inverse of the function that is also a function. Looking at the graph of our function \( f(x)=\frac{x^{3}}{2} \), we can see that each horizontal line drawn through the graph will intersect at only one point.
03

Conclude about the invertibility of the function

Given that the function passes the horizontal line test, we can conclude that the function is one-to-one, which means it has an inverse that is also a function.

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

You will be developing functions that model given conditions. A company that manufactures bicycles has a tixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, \(C\), as a function of the number of bicycles produced. Then find and interpret

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt{x}+1 $$

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{2}-2 $$

Explain how to derive the slope-intercept form of a line's equation, \(y=m x+b,\) from the point-slope form \(y-y_{1}=m\left(x-x_{1}\right)\)

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)=\sqrt{x+2}-2 $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics 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.