"As-tu fini le livre ?"
Translation:Are you through with the book?
In my opinion, your answer is correct and Duo's is wrong. "Are you through with the book?" better translates to "As-tu fini avec le livre?". Are you through with the book doesn't have anything to do with being finished the book, just that you aren't using it now and someone wants to borrow it. As-tu fini le livre is definitely asking if you've finished (reading) the book.
i confirm what hohenems said, as-tu fini le livre ? is definitely a correct answer.