0

Guest

Courses New

Hi all I have functional equation. Find the function such as f(0)=1/2 and exists a for any x,y f(x+y)=f(x)(a-y)+f(y)(a-x) I found that a=1/2 then f(x)=1/2-x It seems OK, but test failed: f(x+y)=1/2-x-y and f(x)(1/2-y)+f(y)(1/2-x)=1/2-x-y+ 2xy I can't find the mistake (logical? calculation?) Ask for help. Thank you.

Hi all
I have functional equation.
Find the function such as f(0)=1/2 and
exists a for any x,y f(x+y)=f(x)(a-y)+f(y)(a-x)
I found that a=1/2 then f(x)=1/2-x
It seems OK, but test failed:
f(x+y)=1/2-x-y and
f(x)(1/2-y)+f(y)(1/2-x)=1/2-x-y+2xy
I can't find the mistake (logical? calculation?)
Ask for help.
Thank you.

Grade:12th pass

3 Answers

Pavan Kumar
32 Points
5 years ago
hiii, thank you for asking the question man,
                   I don’t completely understand your problem but let me solve the problem and check your solution once again
  f(x+y)=f(x)(a-y)+f(y)(a-x)                        -1
             for  a=1/2 then f(x)=1/2-x
                              also  f(y)=1/2-y
                     substitute in -1
                        we get    f(x+y)=f(x)(1/2-y)+f(y)(1/2-x)
                                                 =(1/2-x)(1/2-y)+(1/2-y)(1/2-x)
                                                  =2(1/2-x)(1/2-y)
                                    f(0)=2*1/2*1/2=1/2
            Hope you got your mistake and correct solution
 


 
George
12 Points
5 years ago
Well, if the f(x)=1/2-x is the correct answer, then the left side of equation
f(x+y)=f(x)(1/2-y)+f(y)(1/2-x)
will be f(x+y)=1/2-x-y, but the right side
f(x)(1/2-y)+f(y)(1/2-x)=(1/2-x)(1/2-y)+(1/2-y)(1/2-x)=1/2-x-y+2xy
Pavan Kumar
32 Points
5 years ago
Hey ,George at first i thought something else was your problem...this question has some many errors 
             
                                   f(x+y)=f(x)(a-y)+f(y)(a-x)   
                    put y=0 gives
                                        f(x)=f(x)(a)+f(0)(a-x)
                                                  given f(0)=1/2 substitute it
                                                f(x)=f(x)(a)+1/2(a-x)    
                                          so f(x)=    1/2*(a-x)/(1-a)   
now here comes something wierd      f(x+y)=(a-x)(a-y)/(1-a)
                                                              put y=a
                                                              f(x+a)=0 this is wrong or else if  x and y not equal to a
                                                             your contradiction so atleast they have to give a not equal to ½ and 1 
                                     see for the remaining conditions this contradiction will not come                                                                             

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

View all Questions »

ASK QUESTION

Get your questions answered by the expert for free

Answer and earn gift vouchers

Win Gift vouchers upto Rs 500/-

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Click Here Know More

Complete Self Study Package designed by Industry Leading Experts

Click Here Know More

Live 1-1 coding classes to unleash the Creator in your Child

Click Here Know More

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Click Here Know More