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For every $x\in \mathbb{R}$, $f(x+6)+f(x-6)=f(x)$ is satisfied. What may be the period of $f(x)$? I tried writing several $f$ values but I couldn't get something like $f(x+T)=f(x)$.

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    Let $f(0) = a, f(6) = a+b$. Now extrapolate and see what comes out. You might find it instructive to start with $a = 0$. – 2017-01-19

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Here is a big hint: apply the identity you have been given to $f(x+6)$ to obtain $f(x+6)=f(x+12)+f(x)$ and then ...

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    thank you , i tried it but couldn't get the answer before but now ok. – 2017-01-19
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It follows then that

$$\begin{align}f(x)&=f(x+6)+f(x-6)\\&=f(x+12)+f(x)+f(x-6)\\\implies0&=f(x+12)+f(x-6)\\\implies f(x+12)&=-f(x-6)\\\implies f(x+18)&=-f(x)\\\implies f(x+36)&=-f(x+18)=f(x)\end{align}$$

$$f(x+36)=f(x)$$

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    @matbaz no problem :D – 2017-01-19
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If $f(x+6)+f(x-6)=f(x),$ then $f(x+12)+f(x)=f(x+6)$ and therefore $f(x+12)=-f(x-6)=-(-f(x-24)).$ Hence you have $$f(x)=f(x-36).$$