Theorem: Consider the function y(x) and its inverse function x(y). Let ϵy denote the elasticity of the function y(x) and ϵx denote the elasticity of the inverse function x(y). Then ϵx=1ϵy.

Proof:
ϵx=x(y)yx(y)=1y(x)yx(y)=1y(x)y(x)x=1ϵy.