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# Find all functions watch

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1. 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)).
2. 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 .

~~Simba

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Updated: July 18, 2008
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