Inverse Mapping/Examples/Real Cube Function

Examples of Inverse Mappings
Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:
 * $\forall x \in \R: \map f x = x^3$

The inverse of $f$ is:


 * $\forall y \in \R: \map {f^{-1} } y = \sqrt [3] y$

Proof
From Bijection Example: Real Cube Function $f$ is a bijection.

By definition of the cube root:


 * $\sqrt [3] y := \set {x \in \R: x^3 = y}$

From Inverse Mapping is Bijection, it follows that $f^{-1}$ is likewise a bijection.