Functional Equation Problem | SMO, 2013 - Problem19

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Try this beautiful problem from Singapore Mathematics Olympiad based on Functional Equation.

Problem – Functional Equation (SMO Exam)

Let f and g be functions such that for all real numbers x and y,

\( g (f (x+y)) = f( x ) + (x+y) g (y)\).

Find the value of \( g(0) + g (1) + ……………………+ g (2013) \)

Functional Equation

Funcion

Arbitrary Numbers

But try the problem first…

Answer: 0

Source

First Hint

We can start this problem by considering y = -x.

Then \( g (f (0) ) = f (x) \) for all x. This \(f\) is is a constant function ; namely

\( f (x) = c \) for some c.

Try the rest of the sum ……………………………………………………

Final Hint

For all value of x,y we have

\( (x+y) g(y) = g(f(x+y)) – f(x) = g(c) – c = 0 \)

Since x + y is arbitrary , we must have \( g (y) = 0 \) for all y .Hence

\( g (0) + g ( 1 ) + …………………………..+ g(2013) = 0 \) (Answer).