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Find all functions watch

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    Find all functions f : R --- R such that f(2007) = 2008 and for each x, y in R

    f(4xy) = 2y(f(x + y) + f(x − y)).

    First off, as with all functional equations, you should try x = 0 and y = 0 to see what happens.

    Next note that f(4xy) = f(4yx) and hence:

    2y[f(x + y) + f(x - y)] = 2x[f(x + y) + f(y - x)]

    Now, combined with your knowledge from x = 0, you can form a functional equation that can be solved relatively straight forwardly. You will need to use an appropriate substitution to bring in the information that f(2007) = 2008.

    I hope this is enough to get you started .

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Updated: July 18, 2008

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