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