Question

Let $f\left(x\right)$ be a function satisfying $f\left(x\right)+f(x+2)=20\forall x\in R$, then

• A. $f\left(x\right)$ is a periodic function
• B. ${\int }_{-3}^{13}f\left(x\right)dx=80$
• C. ${\int }_{-1}^{7}f\left(x\right)dx=80$
• D. $f\left(x\right)$ is a many one function

Answer 1

Answer: (A), (C), (D)

$f\left(x\right)$ is a many one function

$f\left(x\right)+f(x+2)=20x\to x+2f(x+2)+f(x+4)=20$
Subtracting
$\Rightarrow$$f\left(x\right)=f(x+4)$
$\Rightarrow$$f\left(x\right)$ is a periodic with a period = 4
${\int }_{-1}^{7}f\left(x\right)dx=2{\int }_0^{4}f\left(x\right)dx=2\left[{\int }_0^{2}f\right(x)dx+{\int }_2^{4}f(x\left)dx\right]=2\left[{\int }_0^2\right(f\left(x\right)+f(x+2)\left)dx\right]=2\left[{\int }_0^{2}20dx\right]=80$
The periodic functions are always many one
${\int }_{-3}^{13}f\left(x\right)dx=4{\int }_0^{4}f\left(x\right)dx=160$