Odd Power Function is Surjective
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Theorem
Let $n \in \Z_{\ge 0}$ be an odd positive integer.
Let $f_n: \R \to \R$ be the real function defined as:
- $\map {f_n} x = x^n$
Then $f_n$ is a surjection.
Proof
From Existence of Positive Root of Positive Real Number we have that:
- $\forall x \in \R_{\ge 0}: \exists y \in \R: y^n = x$
From Power of Ring Negative:
- $\paren {-x}^n = -\paren {x^n}$
and so:
- $\forall x \in \R_{\le 0}: \exists y \in \R: y^n = x$
Thus:
- $\forall x \in \R: \exists y \in \R: y^n = x$
and so $f_n$ is a surjection.
Hence the result.
$\blacksquare$