Find the domain and range of `f(x)=cos^(-1)sqrt((log)_([x])((|x|)/x))`

Answer

Domain: `[2,oo),` Range: `{pi//2}`

Explanation

` "log"_([x])(|x|)/(x)` is defined if
`(|x|)/(x) gt 0, [x] gt 0, " and " [x] ne 1`
or `x gt 0,x in [1,oo), " and " x notin [1,2)`
or `x in [2,oo)`
For `x in [2,oo)` we have `log_([x])(|x|)/(x)=log_([x])1=0.`
Therefore, `f(x)=cos^(-1)0=(pi)/(2)` for all ` x in [2,oo).`
Hence, domain `(f)` is `[2,oo)` and range `(f)` is `{pi//2}.`