My Math Forum  

Go Back   My Math Forum > Math Forums > Math Events

Math Events Math Events, Competitions, Meetups - Local, Regional, State, National, International


Reply
 
LinkBack Thread Tools Display Modes
May 27th, 2013, 09:50 AM   #1
Newbie
 
Joined: May 2013

Posts: 6
Thanks: 0

Functional equation

Find all functions f:R->R such that
frobenius is offline  
 
May 29th, 2013, 08:53 AM   #2
Newbie
 
Joined: May 2013

Posts: 6
Thanks: 0

Re: Functional equation

A couple of simple observation:
f(x)=0 iff x=0
( so we must face both cases).
f(x)=0 is a solution.
frobenius is offline  
May 29th, 2013, 07:28 PM   #3
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,882
Thanks: 1088

Math Focus: Elementary mathematics and beyond
Re: Functional equation

f(x) = x
greg1313 is offline  
May 29th, 2013, 09:46 PM   #4
Newbie
 
Joined: May 2013

Posts: 6
Thanks: 0

Re: Functional equation

Well also f(x)=-x is a solution (one per each case). The difficult is to prove that f is surjective, once done you just need to put x=-1 and it's done.
frobenius is offline  
June 6th, 2013, 05:15 AM   #5
Member
 
Joined: May 2013
From: Phnom Penh, Cambodia

Posts: 77
Thanks: 0

Re: Functional equation

First, we will prove that f is injective.
We have, f(x+f(x)f(y))=f(x)+xf(y).
Let y=f(x), we obtain f(x+f(x)f(y))=y+xf(f(x))
x=0, f(f(0)f(y))=y, then f(f(y))=y/f(0).
If f(y1)=f(y2), so f(f(y1))=f(f(y2))
imply that y1=y2. Hence, f is injective.
since, f(f(0)f(y))=y=f(x).
Then, f(0)f(y)=x
f(y)=x/f(0), f(x)=y.
Let y=1, multiply f(x)f(1)=x/f(0)
f(x)=x/f(0)f(1). Therefore, f(x)=ax, a is a real number.
But this is not satisfy with x=0.
KyVanchhay is offline  
Reply

  My Math Forum > Math Forums > Math Events

Tags
equation, functional



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
A Functional Equation Pell's fish Real Analysis 2 February 3rd, 2014 04:08 AM
functional equation nukem4111 Real Analysis 1 September 16th, 2013 04:44 AM
Functional equation Nobody1111 Real Analysis 7 March 1st, 2012 11:01 AM
Functional equation: x + 2*y + z = x @ (y @ z) antros Applied Math 4 October 12th, 2008 12:12 PM
another functional equation desum Elementary Math 0 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.