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Question
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
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]?
Solution
It is given that,
f: R → R is defined as f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
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 ∈ (0, 1].
Then, we have:
[x] = 1 if x = 1 and [x] = 0 if 0 < x < 1
:. fog(x) = `f(g(x)) = f([x]) = {(f(1), "if x = 1"),(f(0), "if x ∈(0,1)"):}` = `{(1, "if x = 1"), (0, "if x ∈ (0,1)"):}
`gof(x) = g(f(x))`
= g(1) [x > 0]
= [1] = 1
Thus, when x ∈ (0, 1), we have fog(x) = 0and gof (x) = 1.
Hence, fog and gof do not coincide in (0, 1].
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