Injection/Examples/Cube Function

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
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): one-to-one function (one-to-one mapping, one-to-one map)
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