Question

state the largest domain of each functions
f:x=1/1-x and g:x=x-1/x
find the composite funcions
fog, gof^-1
where fog means g followed by f and f^-1 denotes the inverse function of f

Answer 1

(f◦g)(x) = f(g) = 1/(1-g) = 1/(1 - (x-1)/x) = x
(g◦f)(x) = g(f) = (f-1)/f = (1/(1-x) - 1)/(1/(1-x)) = x
note that this means that f and g are inverses of each other

Answer 2

fog: The largest domain of fog is the set of all real numbers except 1.

gof^-1: The largest domain of gof^-1 is the set of all real numbers except 0.