Question

Let $f(x)$ be a continuous function in R such that $f(x)+f(y)=f(x+y)$. If ${\int }_0^{3}f(x)dx=$K, then ${\int }_{-3}^{3}f(x)dx$ is equal to

• A. 2 K
• B. 0
• C. $\frac{K}{2}$
• D. -2K

Answer 1

Answer: (A)

Since $f\left(x\right)+f\left(y\right)=f\left(x+y\right)...\left(1\right)$
$\therefore f\left(x\right)+f\left(-x\right)=f\left(0\right)\dots \left(2\right)$
[By putting $y=-x$ in $(1)]$
Again $f\left(0\right)+f\left(0\right)=f\left(0\right)$
[By putting $x=y=0$ in $(1)$]
$\Rightarrow f\left(0\right)=0\therefore \left(2\right)$ gives
$f\left(x\right)+f\left(-x\right)=0\therefore f\left(-x\right)=-f\left(x\right)$
$\therefore f\left(x\right)$ is an odd function.
$\therefore \underset{-3}{\overset{3}{\int }}f\left(x\right)dx=0$