1.

Find the domain and range of the function given by `f(x)=1/(sqrt(x-[x]))`

Answer» We have, `f(x)=(1)/(sqrt(x-[x]))`
We know that
`0lex-[x]lt1"for all "x""inR`
and `x-[x]=0"for all "x""inZ`.
`:.0ltx-[x]lt1" for all "x""inR-Z`
`impliesf(x)=(1)/(sqrt(x-[x]))` exists for all `x""inR-Z`
`implies"dom "(f)=R-Z`.
Also, `0ltx-[x]lt1 "for all "x""inR-Z`
`implies0ltsqrt(x-[x])lt1 "for all "x""inR-Z`
`implies1lt(1)/(sqrt(x-[x]))ltoo "for all "x""inR-Z`
`implies1ltf(x)ltoo" for all "x""inR-Z`
`implies"range "(f)=(1,oo)`.
Hence, dom `(f)=R-Z" and range "(f)=(1,oo)`.


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