/*! 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 101 The function \(f(x)=k\left(2-x-x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 101

The function \(f(x)=k\left(2-x-x^{3}\right)\) has an inverse function, and \(f^{-1}(3)=-2 .\) Find \(k\).

Short Answer

Expert verified
The value of \(k\) is 0.25.

Step by step solution

01

Write the equation according to the information provided

Given that \(f^{-1}(3) = -2\), we can infer that \(f(-2) = 3\). Now substitute \(x = -2\) into \(f(x)\) and make it equal to 3. So we have: \(3 = k(2 - (-2) - (-2)^3)\)
02

Simplify the equation

In the above equation, calculate the values inside the parenthesis. This becomes: \(3 = k(2 + 2 - (-8)) = k(2 + 2 + 8) = k(12)\)
03

Solve for k

The equation \(3 = k(12)\) can be simplified to: \(k = 3 / 12 = 0.25\) by divide both sides by 12.

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

The table shows the numbers of tax returns (in millions) made through e-file from 2003 through \(2010 .\) Let \(f(t)\) represent the number of tax returns made through e-file in the year \(t .\) (Source: Internal Revenue Service) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number of Tax Returns Made Through E-File } \\\\\hline 2003 & 52.9 \\\2004 & 61.5 \\\2005 & 68.5 \\\2006 & 73.3 \\\2007 & 80.0 \\\2008 & 89.9 \\\2009 & 95.0 \\\2010 & 98.7 \\\\\hline\end{array}$$ (a) Find \(\frac{f(2010)-f(2003)}{2010-2003}\) and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let \(N\) represent the number of tax returns made through e-file and let \(t=3\) correspond to 2003 (d) Use the model found in part (c) to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline N & & & & & & & & \\ \hline\end{array}$$ (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let \(x=3\) correspond to \(2003 .\) How does the model you found in part (c) compare with the model given by the graphing utility?

Sketch the graph of the function. $$g(x)=[[x]]-1$$

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(y\) varies inversely as \(x .(y=3 \text { when } x=25 .)\)

Determine whether the statement is true or false. Justify your answer. The set of ordered pairs \(\\{(-8,-2),(-6,0),(-4,0)\) \((-2,2),(0,4),(2,-2)\\}\) represents a function.

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$k(x)=1 /(x-3)$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied 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.