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The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$f(x) f(y) = f(x + y) + xy$$ for all real numbers $x$ and $y$. Find all possible functions $f$.

Amit Rajaraman
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littlemix
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    Welcome to Stackexchange. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – sai-kartik May 23 '20 at 04:50
  • One solution is 1+x – Dhanvi Sreenivasan May 23 '20 at 04:51
  • Why don't you add some context, like where did that come from, maybe a book or a teacher, or you just invented it yourself, taking some property of some function to investigate whether that identifies it? And why don't you add some background, so we can know what answer you can or cannot digest? Given it's a rather simple case, and you didn't even try anything, you must be a newby in functional equations. –  May 23 '20 at 10:29

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I cannot write the whole answer. Here are hints for you.

Compute $f(0)$ by taking $x=y=0$ If $f(0)=0$ then choose $y=0$... If $f(0)=1$ then find $a$ such that $f(a)=0$ ( $a \in \{1,-1\}$). Then choose $y=a$ deduces the results.

Thanks for the comments. There are 2 solutions.

N.Quy
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