Let $$f\left( x \right)$$  be a continuous function in $$R$$ such that $$f\left( x \right) + f\left( y \right) = f\left( {x + y} \right).$$     If $$\int_0^3 {f\left( x \right)dx = k} $$    then $$\int_{ - 3}^3 {f\left( x \right)dx} $$    is equal to

Solution :
$$\eqalign{ & {\text{Putting }}x = 0,\,y = 0,\,{\text{we get}}\,{\text{ }}f\left( 0 \right) + f\left( 0 \right) = f\left( 0 \right) \cr & \therefore f\left( 0 \right) = 0 \cr & {\text{Putting }}y = - x,\,f\left( x \right) + f\left( { - x} \right) = f\left( {x - x} \right) = f\left( 0 \right) = 0 \cr & \therefore f\left( { - x} \right) = - f\left( x \right) \cr & \therefore f\left( x \right)\,{\text{is an odd function}}{\text{. So, }}\int_{ - 3}^3 {f\left( x \right)dx = 0} \cr} $$