Inverse Mapping/Examples/Real Cube Function
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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