Question

Let $f$ : $R\to R$ be a continuous function given by $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y\in R$ lf ${\int }_0^{2}f\left(x\right)dx=\alpha$, then ${\int }_{-2}^{2}f\left(x\right)dx$ is equal to

• A. $2\alpha$
• B. $\alpha$
• C. 0
• D. $\alpha -2$

Answer 1

Answer: (C)

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 odd function
Gives ${\int }_{-2}^{2}f\left(x\right)dx=0$