We proceed using a direct proof. Assume is invertible. Then is one-to-one and onto (there is a theorem that tells us this, it should be in your text).
We show is onto: Let . Since is onto, there exists such that . But that means with . Thus is onto.
We show is one-to-one: Assume with . Then , or in other words, . But is one-to-one, so . Hence, is one-to-one