Odd Power Function is Surjective

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 we have that $\left({-x}\right)^n = - \left({x^n}\right)$ 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.