# 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:

$\forall x \in \R_{\ge 0}: \exists y \in \R: y^n = x$
$\paren {-x}^n = -\paren {x^n}$

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.

$\blacksquare$