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Let g(x)=1=x-[x] and f(x)={(-1",",x lt 0...

Let `g(x)=1=x-[x]` and `f(x)={(-1",",x lt 0),(0 ",",x =0),(1",",x gt 0):}` then for all `x, f(g(x))` is equal to
(i) `x`
(ii) `1`
(iii) `f(x)`
(iv) `g(x)`

Text Solution

Verified by Experts

The correct Answer is:
`f(g(x))=1` for all `x`
We have `g(x)=1+x-[x]=1+{x},` where `{x}` represent fractional part function
and `f(x)={(-1",",x lt 0),(0 ",",x =0),(1",",x gt 0):}`
`implies f(g(x))={(-1",",1+{x} lt 0),(0 ",",1+{x} =0),(1",",1+{x} gt 0):}`
`implies f(g(x))=1,1+{x} gt 0 " " ( :' 0 le {x} lt 1)`
`implies f(g(x))=1 AA x in R`
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