Let take and take . Then , so , and similarly , so . Hence is invertible with inverse .

A left inverse alone does not make a function invertible. Let take and take . Then , but since . Both composites are required.

is invertible if and only if is bijective, and the inverse is written .

Suppose some satisfies . If , then , so must be injective. Conversely, let be injective: for set for the unique with , and for choose arbitrarily in ; then . For as well, surjectivity is necessary because forces onto , and sufficient because each equals for some , so taking gives . The construction of is consistent across the two parts, so the result follows.