/*! 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} Q 107E Question 107: Let 鈩� be the set... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Q 107E (page 105)

Question 107: Let 鈩� be the set of all positive integers, 1, 2, 3,鈥�. We define two functions f and g from鈩� to 鈩�:

f(x)=2x, for all x in 鈩�

g(x)={x2鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌塱蹿鈥�xisevenx+12鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌塱蹿xisodd

Find formulas for the composite functionsg(fx) and f(gx). Is one of them the identity transformation from鈩� to 鈩�? Are the functions fand g invertible?

Short Answer

Expert verified

The formulas aregfx=x , fgx=x鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌�,xisevenx+1鈥�,xisodd,gfx is an identity transformation. The functions fand g are invertible.

Step by step solution

01

Write the given data:

Given two functions f and g from鈩� to 鈩�:

fx=2x,for all x in鈩�

gx=x2鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌塱蹿鈥�xisevenx+12鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌塱蹿xisodd

02

Find  g ( f ( x ) )

The functionfx=2xis even, solve for the function as:

gfx=g2x=2x2=x

Hence,g ( f ( x ) ) = x for all x.

03

Find  f ( g ( x ) ) 

Let xis even, then

fgx=fx2=2x2=x

If x is odd, then

fgx=fx+12=2x+12=x+1

Thus, fgx=x鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌�,xisevenx+1鈥�,xisodd

04

Check identity transformation

Since,g ( f ( x ) ) = x for all x, sog ( f ( x ) ) is an identity transformation.

05

Check whether f and g are invertible

Now, since f鈭�1x=x2

And g鈭�1x=2x鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆�夆赌夆赌夆�夆赌夆赌�,xiseven2x+1鈥夆赌夆赌�,鈥�xisodd

Therefore, f and g are invertible.

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

ifA2=I2 then matrix A must be eitherI2or-I2 .

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises 16through 23 . Show the effect of these transformations on the letter L considered in Example 5 . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.

21. [01-10]

TRUE OR FALSE?

If A2=In, then matrixAmust be invertible.

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation T(x)鈫�=Ax鈫�on this face.

29.[-100-1]

TRUE OR FALSE?

The matrix(111-1) represents a reflection about a line.

See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.