Question

f(x)=the squareroot of (x^2-9) and
g(x)=1/x-4
find the domain and range of f(g(x)) anf f(f(x))

Answer 1

domain of f is all reals
domain of g is all reals except x=4

f(g) = g^2 - 9 = 1/(x-4)^2 - 9
domain of f(g) is the domain of g
range is all reals > -9

f(f) = f^2 - 9 = (x^2-9)^2 - 9
domain and range are all reals.

I expect you meant to type g(f(x)), but I'll let you do that as penalty...