# Problem of the day 3

Nice problem !

Functional Equations !

Q)
Determine all functions ƒ satisfying  the functional relations

ƒ(x) + ƒ(1/(1-x)) =( 2(1-2x)/(x(1-x))

where x is a real number x ≠ 0 , x ≠ 1

person who give best answer of problem of the day more than 4 times will be awarded a book

I am a young student who loves math . I like number theory and inequalities part the most , and preparing for Math Olympiads :)

Posted on Saturday,June 9, 2012, in Problems and tagged , , , , , , . Bookmark the permalink. 6 Comments.

1. Ans : f(x) = (x 1)/(x-1)

solution :

f(x) f(1/1-x) = 2/x – 2/(1-x) ….1

substitute (1/1-x) for x in the above and get the eqn,

f(1/1-x) f((x-1)/x) = 2/x – 2x …..2

again, substitute 1/1-x in the above and get,

f((x-1)/x) f(x) = 2(1-x) – 2/(1-x) …..3

subtract 2nd equation from 3rd and add the resultant equation to the 1st equation.
Ultimately we have,
f(x) = (x 1)/(x-1)

2. 🙂

3. Ans: f(x)=(x+1)/(x-1)

Detailed solution is as follows

Given:

ƒ(x) + ƒ(1/(1-x)) =( 2(1-2x)/(x(1-x)) …(1)

Substituting x as 1/(1-x) , we obtain

f(1/(1-x)) + f( (x-1)/x) = 2(1- x^2)/x ….(2)

Substituting further x as (x-1)/x in (1) , we get

f(x) + f( (x-1)/x) = 2(x-2)x/(1-x) ….(3)

Adding (1), (2), and (3), dividing by 2 and subtracting (2), we obtain

f(x)=(x+1)/(x-1).

4. keshushivang

Saikat :

Ans : f(x) = (x 1)/(x-1)

solution :

f(x) f(1/1-x) = 2/x – 2/(1-x) ….1

substitute (1/1-x) for x in the above and get the eqn,

f(1/1-x) f((x-1)/x) = 2/x – 2x …..2

again, substitute 1/1-x in the above and get,

f((x-1)/x) f(x) = 2(1-x) – 2/(1-x) …..3

subtract 2nd equation from 3rd and add the resultant equation to the 1st equation.
Ultimately we have,
f(x) = (x 1)/(x-1)

Good Saiket , you are absolutely correct 🙂

5. priyanshi

Can u send some now routine problems also.. I m preparing for nmtc final contest.. and ur website seems itnteresting

6. You can freely chat with me here : https://www.facebook.com/si8ma007
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