january(1-15) 2009 maths

find all functions f:R⁺â†’R⁺ such that

f(x+f(y))=f(x+y)+f(y) for all x,y belonging to R⁺

3 Answers

1 fn is.. y=2x

another is f(x)=0,

matlab kisi ko aur koi nahi mila... that means these are the only one..[3]

341

Hari Shankar ·2009-02-12 06:10:43

We can prove that no other solutions exist. Let f(x) = 2x+g(x) be a solution to the given functional equation.

Then LHS = f(x+f(y)) = f(x+2y+g(y)) = 2(x+2y)+2g(y) + g(g(y))

RHS = f(x+y) + f(y) = 2(x+y)+g(x+y)+2y+g(y) = 2(x+2y)+g(x+y)+g(y)

This implies that g(x+y) = g(y) + g(g(y) is independent of x

In other words g(x) is a constant = c. Putting it back in the equation above, we have c = 0

Hence f(x) = 2x is the unique solution satisfying the given conditions

f(x) = 0 doesnt qualify as it is given f(x):R⁺â†’R⁺

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