Find the domain of definition of the following functions :
(i) `f(x) = log_(10) sin (x-3) + sqrt(16 - x^(2))`
(ii) `f(x) = sqrt((4- |x|)/(7-|x|))`
(iii) `f(x) = (1)/(sqrt([|x|-1]|-5))` where [x] denotes the greatest integer function.

Explanation

(i) `f(x) = log_(10) sin(x-3) + sqrt(16 - x^(2))`
`= f_(1) (x) + f_(2)(x)` (say)
`f_(1)(x)` is defined if `sin(x-3) gt 0, 2n pi lt x x - 3 lt 2 n pi + pi, n in Z` (general solution)
`2n pi + 3 lt x lt 2n pi + pi + 3, n in Z`.
For `f_(2)(x)` to be defined `16 - x^(2) gt 0`
`x^(2) -1 6 le 0`
`:. D_(f) = D_(f_(1)) cap D_(f_(2)) x in [-4, 4]`
`= (-2pi + 3, -pi + 3) cup (3, 4]`
(ii) `f(x) = sqrt((4-|x|)/(7-|x|))`
f(x) is defined (f(x) is a finite real number)
iff `(4-|x|)/(7-|x|) ge 0` and `|x| != 7`
iff `(|x| - 4) (|x| - 7) ge 0` and `|x| != 7`
iff `|x| le 0` or `|x| gt 7`
iff `x in [-4, 4]` or `x lt -7` or `x gt 7`
`:. D_(f) = [-4, 4] cup (-oo, -7) cup (7, oo)`
(iii) `f(x) = (1)/(sqrt(|[|x| - 1]| - 5))`
f(x) is defined (f(x) is a finite real number)
iff `|[|x| - 1]| - 5 gt 0`
iff `|[|x| - 1]| gt 5`
iff `[|x|- 1] lt -5` or `[|x| - 1] gt 5`
iff `|x| - 1 lt -5` or `|x| - 1 ge 6`
iff `|x| lt -4` or `|x| ge 7`
iff (not possible) or `x le -7` or `x le 7 " " :. D_(f) = (-oo, -7) cup [7, oo)`