I did this pretty quickly but I think this will hopefully help you,
then multiplying all by exp(x) gives
and since y = cosh(x) = cosh( ln( y .... ) ) then
Now, ln(a) is a real valued function where a must be positive and real, so y +/- sqrt(y^2 - 1) must be positive and real; thus y^2 - 1 >= 0 => y >= 1 (which is true since by  that x >= 0 and so cosh(x) >= 1), now for y +/- sqrt(y^2 - 1) to be positive then we must take y + sqrt(y^2 - 1)
which (for letting y be x) is what was wanted.
The domain for arccosh(y) would then be such that y + sqrt(y^2 - 1) >= ie y > 1 (since ln(0) is also undefined) and the range would then be all positive real numbers (from 0 to +infinity) which is the range of ln for domain greater than 1.