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Functional Equation Problem | SMO, 2013 - Problem19 (Senior Section)

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 $g(0) + g (1) + ........................+ g (2013)$

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Key Concepts

Functional Equation

Funcion

Arbitrary Numbers

Challenges and thrills

Try with Hints

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 ............................................................

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).