Range/Domain Problem

so question goes something like this ...

f(x) = 1/(1+ln(x))

g(x) = 1-sin(1+x^2)

and i am asked for the range of f^-1(g(y))

i worked out f^-1(g(y)) to be: e^((1/g(y))-1)

now this is defined provided the range of g(y) is a subset of the Domain of f^-1

for this to be true, the Range of g(y) must be restricted to (0, 2] {as opposed to the [0,2] it was before}

now this is the step i do not understand... my answer book says that therefor the range of f^-1(g(y)) is [e^-1/2,∞)

can any1 please explain to me the last step?

thanks