## If f(a+b) = f(a) + f(b) + 2ab, what is f?

December 1, 2006
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OK, I think I have a proof of the following:

If , then for some real m.

Proof:

Setting and , we have ;

in other words .

Next, set and , which yields .

Since , we have , or

.

Let (the left side)

Then we see that and is an odd function, i.e. .

Now,

but also

Setting equal the right sides of the above two equations and simplifying,

we get

,

which is the definition of linear function with . (y-intercept = 0)

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Categories: math