Question

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

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

Answer 1

The correct option is B $0$

Substituting $y=0$ in $f(x+y)=f\left(x\right)+f\left(y\right)$, we get,

$f\left(x\right)=f\left(x\right)+f\left(0\right)\Rightarrow f\left(0\right)=0$
$f\left(x\right)+f(-x)=f(x+(-x))=f\left(0\right)=0$
Hence, $f\left(x\right)$ is an odd function.
This gives ${\int }_{-3}^{3}f\left(x\right)dx=0$