Doubt from functions

Question no 1 part f shouldn't be the domain be (-infinity,-2] union [6,infinity] instead of given answer

I am getting |x| belongs to (-infinity, -2] U [6, infinity). Since, mod x can't be negative we get.

|x| belongs to [6, infinity) or x belongs to (insert the given answer here).

Ya it means I am right the answer key is wrong

No, the answer key is correct. Domain is the set of values of x and not |x|, so x belongs to (-infinity, -6] U [6, infinity).

Why in example 1.95 and 1.98 third log term is eliminated while finding domain of function

like log base 10 term eliminated in 1.95 and log base 4 eliminated in 1.98

For log base 10 to be defined, its argument should be greater than zero. That's what the step is done.

The argument is made greater than zero.