composite functions

Question

let f(x)=sqrtx-1 and g(x)=1/x

find f(f(x))

find the domain of (g(f(x))

Answer 1

f (f (x)) = sqrt (f(x)) - 1 = sqrt (sqrt(x) - 1) - 1.

 

g( f(x) ) = 1 / [sqrtx-1].  Assuming that f(x) is real-valued, x must be non-negative; so x >= 0. 

 

We can not have 0 in the denominator of g(f(x)).  Hence the domain of g(x)  is   0 =< x < 1 and 1 < x =< +infinity.