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# Functional Equation

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The function $$f : \mathbb{R} \to \mathbb{R}$$ satisfies

$$f(x) f(y) = f(x + y) + xy$$

for real numbers $$x$$ and $$y$$.. Find all possible functions $$f$$.

I've found that $$f(0)=1$$ and if $$f(-1)$$ is not $$0,$$ then $$f(1)=0.$$ I'm just stuck and any answer/help would be appreciated! (btw i got the values for $$f(0)$$ and $$f(1)$$ by plugging in values for x and y like 0 and 0, or x=1, y=1.)

------------Thanks!

Jul 12, 2020
edited by madyl  Jul 12, 2020

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I checked if there were constant solutions, and there are none, so I don't think there are any other solutions.

Jul 12, 2020
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What do you mean checked constant solutions, im pretty sure there is a solution so can you show me your work?

madyl  Jul 12, 2020
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How about  f(x) = x + 1

f(x).f(y) = x + y + x.y + 1

f(x+y) + x.y = x + y + x.y + 1

Jul 13, 2020