/*! 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 32 Find the inverse of \(f .\) Then... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 32

Find the inverse of \(f .\) Then use a graphing utility to plot the graphs of \(f\) and \(f^{-1}\) using the same viewing window. $$ f(x)=\frac{x}{\sqrt{x^{2}+1}}, \quad-1 \leq x \leq 1 $$

Short Answer

Expert verified
The inverse function is \(f^{-1}(x)=\frac{x}{\sqrt{x^2-1}}\). To plot the graphs of \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\) and its inverse, use a graphing utility and set the viewing window to include the range \([-1, 1]\) for both x and y. You'll notice that the graphs are mirror images of each other with respect to the line y=x.

Step by step solution

01

Rewrite the function with y

Rewrite the given function \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\) as \(y=\frac{x}{\sqrt{x^{2}+1}}\).
02

Switch the roles of x and y

Now, replace x with y and y with x to get: \(x=\frac{y}{\sqrt{y^{2}+1}}\)
03

Solve for y

To solve for y in the expression \(x=\frac{y}{\sqrt{y^{2}+1}}\), follow these steps: 1. Multiply both sides by \(\sqrt{y^{2}+1}\), which gives us: \(x\sqrt{y^{2}+1} = y\). 2. Square both sides to get rid of the square root, which results in: \(x^{2}(y^{2}+1) = y^{2}\). 3. Expand the left side: \(x^2y^2+x^2=y^2\). 4. Rearrange as a quadratic equation in y: \((x^2-1)y^2+x^2=0\). 5. Factor: \(y^2(x^2-1)+x^2=0\). 6. Solve for y^2: \(y^2=\frac{x^2}{x^2-1}\). 7. Finally, we'll consider only the positive square root because the function is continuous: \(y=\frac{x}{\sqrt{x^2-1}}\). Thus, the inverse function is \(f^{-1}(x)=\frac{x}{\sqrt{x^2-1}}\).
04

Plot the graphs of f and \(f^{-1}\) using the same viewing window

To plot the graphs of f and its inverse, we can use any graphing utility such as Desmos, GeoGebra, or a graphing calculator. Set the viewing window to include the range [-1, 1] for both x and y to cover the domain and range of the given function and its inverse. Enter the function \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\) and its inverse \(f^{-1}(x)=\frac{x}{\sqrt{x^2-1}}\) in the graphing utility and observe the graphs on the same viewing window. You'll notice that the graphs of the original function and the inverse are mirror images of each other with respect to the line y=x, which is a property of inverse functions.

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

Write the expression in algebraic form. $$ \tan \left(\tan ^{-1} x\right) $$

Let \(f(x)=x+\frac{1}{100} \sin 100 x\) a. Plot the graph of \(f\) using the viewing window \([-10,10] \times[-10,10]\) b. Plot the graph of \(f\) using the viewing window \([-0.1,0.1] \times[-0.1,0.1]\) c. Explain why the two displays obtained in parts (a) and (b) taken together give a complete description of the graph of \(f\).

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=x^{2} \sin \frac{1}{x} $$

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\cos x, \quad y=\frac{1}{2} \cos \left(x-\frac{\pi}{4}\right)\)

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=\sqrt{x}+1\); shifted horizontally to the left by 1 unit, compressed horizontally by a factor of 3, stretched vertically by a factor of 3, and shifted vertically downward by 2 units
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

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.