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:

$\forall y \in \R: \inv f y = \sqrt [3] y$

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.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions