The domain and range of `f(x) = cos^(-1)sqrt(log_([x])(|x|/(x)))` where[.] denote the greatest integer function are respectively

Options

A. `[1, oo), [0, pi//2]`
B. `[2, oo), [0, pi//2)`
C. `[2, oo), {pi//2}`
D. `[1, oo), {0}`

Answer

C

Explanation

We observe that `log_([x]).(|x|)/(x)` is defined, if
`(|x|)/(x) gt 0 ,[x] gt 0 nd [x] ne 1`
`rArr x gt 0, x in [1,oo) and x notin [1,2) rArr x in [2,oo)`
For `x in [2,oo)` , we find that
`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) = [2,oo)` and range `(f) ={(pi)/(2)}`