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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)
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