This statement is true.
If the domain of a function F is the set of all possible inputs, then the range of F is the set of all possible outputs.
If F^-1 exists, it is the inverse function of F, which means that it takes the outputs of F as its inputs and outputs the inputs of F as its outputs.
Since the domain of F is the same as the range of F^-1, and the range of F is the same as the domain of F^-1, it follows that if F^-1 exists, the domain of F is the same as the range of F.