if f(x) = sqrt x^2-1 and g(x)-sqrt x-1, which expression represents f(x)/g(x) for x>1.

You have a couple of typos.

I will assume you meant
f(x) = √(x^2-1) and g(x) = √(x-1)

then f(x)/g(x)
= √( (x^1-1)/(x-1) )
= √( (x+1)(x-1)/(x-1) )
= √(x+1) , for all x > -1

Since you didn't supply the choices they gave you .....

btw, the way you typed it . it would be
(√(x^2) - 1 )/(√x - 1)
= (x-1)/(√x - 1)
which reduces to √x + 1 , where x > 1

notice how important those brackets are ??