Let f(x)= arctan( e^-x ) + 1 ; x belongs to R

1- Find range of f

2- Determine whether is f 1-1 or not.

3- Find inverse of f

My answer:

1- range of f is [1 , (2+pi)/pi ] .. I get it by evaluating limits of f at infinity and -infinity.

2- f'(x) = - e^(-x) / ( e^(-2x) + 1 ) which is < 0 ----> f is decreasing ----> f is 1-1 ----> inverse exists.

3- f^(-1)(x) = - ln ( tan (x-1) )

Are my answers correct?

Thanks