<s>[INST] How to find the value of this function If $f:\mathbb{R} \rightarrow\mathbb{R}$ is a function which satisfies $$f(x)+f(y)=\frac{f(x-y)}{2}\cdot \frac{f(x+y)}{2}$$ for all $x\in\mathbb{R}$ and $f(1)=3$, what is $f(6)$?
 [/INST] The function you described does not exist because no value of $f(0)$ makes sense:


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*$f(0) + f(0) = \frac{f(0 - 0)}{2} \cdot \frac{f(0+0)}{2} = \frac{f(0)^2}4$
implies that $8f(0) = f(0)^2$ which means $f(0)$ can either be $0$ or $8$.

*$3+f(0) = f(1) + f(0) = \frac{f(1-0)}{2}\cdot \frac{f(1+0)}{2} = \frac{f(1)^2}{4} = \frac{9}{4}$ means that $f(0) = \frac{9}{4} - 3 = -\frac34$, which leads to a contradiction.

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