Question

The period of the function $f\left(x\right)$ , which satisfies the relation $f\left(x\right)+f\left(x+4\right)=f\left(x+2\right)+f\left(x+6\right)$ is

Answer 1

Answer: 8

Given that,
$f(x)+f(x+4)=f(x+2)+f(x+6)$
Replacing $x$ with $x+2$ ,
$\Rightarrow f(x+2)+f(x+6)=f(x+4)+f(x+8)$
Adding both above equations,
$\Rightarrow f(x)+f(x+2)+f(x+4)+f(x+6)=f(x+2)+f(x+4)+f(x+6)+f(x+8)$
$\Rightarrow f(x)=f(x+8)$
$\Rightarrow f(x)$ is a periodic function and its period is $8$ .