Find the domain of (a)`f(x)=1/(sqrt(x-[x]))` (b) `f(x)=1/(log[x])` (c)`f(x)=log{x}`

Answer

(a) `R-Z` (b) `[2,oo)` (c ) `R- Z`

Explanation

We have `f(x) =(1)/(sqrt(x=[x]))`
We know that `0 le x -[x] lt1` for all ` x in R.` Also, `x-[x]=0` for `x in Z.`
Now, `f(x)=(1)/(sqrt(x-[x]))` is defined if
`x-[x] gt 0`
or `x in R-Z " "[ :' x -[x]=0 " for " x in Z " and " 0 lt x lt [x] lt 1 " for " x in R-Z]`
Hence, domain`=R-Z.`
(b) `f(x)=(1)/(log[x])`
We must have `[x] gt 0 " and " [x]ne 1 ("as for" [x]=1, log[x]=0).`
Therefore, `[x] ge 2 " or " x in [2,oo).`
(c ) `f(x) = log{x}` is defined if `{x} gt 0` which is true for all real numbers except integers.
Hence, the domain is `R-L`.