Let f : R → R be the Signum Function defined as and g : R → R be the Greatest Integer Function given by g ( x ) = [ x ], where [ x ] is greatest integer less than or equal to x . Then does f o g and g o f coincide in (0, 1]?
It is given that the function f : R → R is Signum function that is defined as,
f ( x ) = { 1 , x > 0 0 , x = 0 − 1 , x < 0
And g : R → R is the Greatest Integer Function defined by,
g ( x ) = [ x ]
Consider that x ∈ ( 0 , 1 ] , then,
[ x ] = 1 if x = 1 and [ x ] = 0 if 0 < x < 1
Now,
f ο g ( x ) = f ( g ( x ) ) = f ( [ x ] ) = { f ( 1 ) , i f x = 1 f ( 0 ) , i f x ∈ ( 0 , 1 ) = { 1 , i f x = 1 0 , i f x ∈ ( 0 , 1 )
Also,
g o f ( x ) = g ( f ( x ) ) = g ( 1 ) = [ 1 ] = 1
It can be observed that when x ∈ ( 0 , 1 ) , f o g ( x ) = 0 and g o f ( x ) = 1 .
Therefore, f o g and g o f do not coincide in ( 0 , 1 ] .
It is given that the function
And
Consider that
Now,
Also,
It can be observed that when
Therefore,
