Question

Let $f\left(x\right)$ be a function satisfying $f\left(x\right)+f(x+2)=10\forall x\in R$, then

• A. $f\left(x\right)$ is a periodic function
• B. $f\left(x\right)$ is aperiodic function
• C. ${\int }_{-1}^{7}f\left(x\right)dx=20$
• D. ${\int }_{-1}^{7}f\left(x\right)dx=40$

Answer 1

The correct option is A $f\left(x\right)$ is a periodic function

Given Functional equation is $f\left(x\right)+f(x+2)=10\cdots \left(1\right)$

Replace $x$ with $x+2$

$⟹f(x+2)+f(x+4)=10\cdots \left(2\right)$

$\left(1\right)-\left(2\right)$

$⟹f\left(x\right)+f(x+2)-f(x+2)-f(x+4)=10-10$

$f\left(x\right)-f(x+4)=0$

$⟹f\left(x\right)=f(x+4)$

Hence the function is periodic function.