Determine f(g(x))

`f(g(x))=sqrt(sqrt(x^2-1)-2)`

The domain of this function is all the numbers excepted 

`x^2-1<0`             (1)  and

`sqrt(x^2-1)-2<0`     (2)

Solve (1) for x

`x^2<1`

`x<1`  and `x>(-1)`

Solve (2) for x

`x^2-1<4`

`x^2<5`

` ` `x<sqrt5` and `x>(-sqrt5)` which includes `-1<x<1`

Thus the domain of the function is all real numbers excepted `-sqrt5<x<sqrt5`

Determine g(f(x)):

`g(f(x))=sqrt((x-2)-1)=sqrt(x-3)`

The domain of this function is all real numbers except x<3.