Show that the function f is bijection

[zodiacbrave]

a function f, that maps from the Cartesian Product of the positive integers to the positive integers. where
f(x,y) = 2^(x - 1) * (2y - 1).

I have to show that this function is both one-to-one and onto. I started trying to prove that it is onto, showing that there exists an n such that f(n,0) = n but I am not sure where to go from here.

Thank you

 

[chiro]

Hey zodiacbrave and welcome to the forum.

For one-to-one, one suggestion I have is to show that the inverse exists everywhere in the respective domain.

By showing that the inverse exists everywhere in the domain, you have basically shown the one-to-one property.

Even though we are only dealing with integers, if you show this property over the positive reals, then it automatically applies for the positive integers (think of it in terms of subsets).

Hint: What do we need for the derivative to be when an inverse function exists across an interval?