/*! 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 10 Find the inverse function of \(f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 10

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=\frac{x-1}{5}$$

Short Answer

Expert verified
The inverse function of \(f(x) = \frac{x - 1}{5}\) is \(f^{-1}(x) = 5x + 1\). Both identities \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) are verified and hold true.

Step by step solution

01

Finding the Inverse Function

First, to find the inverse function, let \(y = f(x)\), then solve for \(x\). In this case, \(y = \frac{x - 1}{5}\) implies that \(x = 5y + 1\). Now, swap \(x\) and \(y\) to get the inverse function: \(f^{-1}(x) = 5x + 1\).
02

Verifying the Identity \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) into \(f(x)\). Thus, \(f(f^{-1}(x))= f(5x + 1) = \frac{((5x + 1)-1)}{5} = x\). So \(f(f^{-1}(x)) = x\) holds true.
03

Verifying the Identity \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) into \(f^{-1}(x)\). Thus, \(f^{-1}(f(x))= f^{-1}(\frac{x - 1}{5}) = 5 * \frac{x - 1}{5} + 1 = x\). So \(f^{-1}(f(x)) = x\) holds true.

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 height \(y\) (in feet) of a baseball thrown by a child is $$y=-\frac{1}{10} x^{2}+3 x+6$$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

Match the data with one of the following functions $$f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \quad \text {and} \quad r(x)=\frac{c}{x}$$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$\begin{array}{|c|c|c|c|c|c|}\hline x & -4 & -1 & 0 & 1 & 4 \\\\\hline y & -32 & -2 & 0 & -2 & -32 \\\\\hline \end{array}$$

The diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. A stream with a velocity of \(\frac{1}{4}\) mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter.

Finding a Mathematical Model In Exercises \(41-50\), find a mathematical model for the verbal statement. Newton's Law of Cooling: The rate of change \(R\) of the temperature of an object is directly proportional to the difference between the temperature \(T\) of the object and the temperature \(T_{e}\) of the environment in which the object is placed.

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=3 x^{2}-1.75$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.