Q. 184.3( 12 Votes )

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 fog and gof coincide in (0, 1]?

Answer :

It is given that

f: R R be the Signum Function defined as



Also, g: R R is defined as g(x) = [x], where [x] is the greatest integer less than or equal to x.


Now, let x ϵ (0, 1]


Then, we get,


[x] = 1 if x = 1 and [x] = 0 if 0 < x < 1


Therefore, fog(x) = f(g(x)) = f([x]) =


gof(x) = g(f(x))


= g(1) [x > 0]


= [1] = 1


Then, when x ϵ (0,1), we get fog(x) = 0 and gof(x) = 1


But fog(1) ≠ gof(1)


Therefore, fog and gof do not coincide in (0, 1].


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