Theorem: Consider the function $y(x)$ and its inverse function $x(y)$. Let $\epsilon^y$ denote the elasticity of the function $y(x)$ and $\epsilon^x$ denote the elasticity of the inverse function $x(y)$. Then $$\epsilon^x=\frac{1}{\epsilon^y}.$$
Proof:
$$\begin{align}
\epsilon^x =& x'(y)\cdot\frac{y}{x(y)}\\
=& \frac{1}{y'(x)} \cdot \frac{y}{x(y)}\\
=& \frac{1}{y'(x)} \cdot \frac{y(x)}{x}\\
=& \frac{1}{\epsilon^y}.
\end{align}$$
Proof:
$$\begin{align}
\epsilon^x =& x'(y)\cdot\frac{y}{x(y)}\\
=& \frac{1}{y'(x)} \cdot \frac{y}{x(y)}\\
=& \frac{1}{y'(x)} \cdot \frac{y(x)}{x}\\
=& \frac{1}{\epsilon^y}.
\end{align}$$