Ques1) Let f(x) = max{x,0} for all x belonging to R and f(xy) is not equals to f(x) f(y), then show that x <0 , y<0.

f(x) can be written as ,

f(x)= o if x<=0

and f(x)=x if x>0

now consider the cases (1)x,y>0  (2) x>0 y<0 or x<0 y>0 (3) x<0 y<0

(1) f(xy) = xy

   f(x)f(y)=x.y=xy

so f(xy)=f(x)f(y)

(2)

f(xy)=0 as xy<0

f(x).f(y) = 0 as f(x)=0 or f(y)=0

(3)

f(xy)=xy as xy>0

f(x)=0 and f(y)=0 => f(x)f(y)=0

therefore f(xy)  is not equal to f(x)f(y)