prove that only one one onto function possess an Inverse function
Answers
we have to prove that , A function is invertible function if and only if it is bijective ( means one to one and onto).
proof : suppose is an invertible function. and is inverse of it.
first we have to show that f is injective (one to one ).
suppose are such that f(a) = f(b)
then we can apply to both sides to get, .
now by the definition of inverse, both the above compositions are the identity on A. so, we can
but we know, if are such that f(a) = f(b) a =b, then f is one-to-one function or injective function.
now, to show surjective function. let and let then,
[ as f is inverse so,
but we know, from definition of surjective, if in such a way that and
then, f is surjective or onto function.
hence, given function, f is also surjective or onto function.
now it is proved that a function is invertible function if and only if it is one to one and onto.