Damn the Functional Equation

Algebra Level 3

A Function is defined as follows -
f : R + → R ; x f ( y ) − y f ( x ) = f ( x y ) ∀ x , y ∈ R \large{f: \mathbb{R}_+ \to \mathbb{R};\; xf(y) - yf(x) = f(\frac{x}{y})}\; \forall x,y \in \mathbb{R}
Find f ( 101 ) f(101) .


The answer is 0.

Rezwan Arefin by Rezwan Arefin , 19, Bangladesh

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1 solution

Rezwan Arefin
May 28, 2016

Substitute x = y x=y to get f ( 1 ) = 0 f(1)=0 .
Then substitute y = 1 y=1
= > x f ( 1 ) − f ( x ) = f ( x ) =>xf(1) - f(x) = f(x)
= > x f ( 1 ) = 2 f ( x ) =>xf(1) = 2f(x)
f ( x ) = 0 f(x) = 0
So f ( 101 ) = 0 f(101) = 0


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