Find the values of x which the function `f(x)=sqrt(log_(1//2)((x-1)/(x+5))` is defined.

Answer

` x in (1, infty)`

Explanation

`f(x)=sqrt(log_(1//2(x-1)/(x+5)))`
It is defined if `log_(1//2).(x-1)/(x+5) ge 0`
` rArr 0 lt(x-1)/(x+5) le 1`
When `(x-1)/(x+5) gt 0, x in (-infty, -5) uu(1, infty)` ...(i)
When `(x-1)/(x+5) le 1`
`rArr (x-1)/(x+5) - 1 le 0`
`rArr (-6)/(x+5) le 0`
` rArr x gt - 5` ...(ii)
From (i) and (ii), ` x in (1, infty)`.