Question

Let $f$ be a function satisfying $f\left(x\right)+f(x+6)=f(x+3)+f(x+9).$ If fundamental period of $f\left(x\right)$ is $T,$ then $T$ equals

Answer 1

$f\left(x\right)+f(x+6)=f(x+3)+f(x+9)...\left(1\right)$
Replace $x$ by $x+3$
$\Rightarrow f(x+3)+f(x+9)=f(x+6)+f(x+12)...\left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we get
$f\left(x\right)+f(x+6)=f(x+6)+f(x+12)$
$\Rightarrow f\left(x\right)=f(x+12)$
$\therefore T=12$