Injection/Examples/Cube Function
Jump to navigation
Jump to search
Example of Injection
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = x^3$
Then $f$ is an injection.
Proof
From Odd Power Function on Real Numbers is Strictly Increasing, $f$ is strictly increasing.
From Strictly Monotone Real Function is Bijective, it follows that $f$ is bijective.
Hence by definition $f$ is an injection.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text I$: Sets and Functions: Composition of Functions
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 22$: Injections; bijections; inverse of a bijection