Cube Function is Odd

Theorem

The cube function on the real numbers:

$\forall x \in \R: \map f x = x^3$

is an odd function.

Proof

\(\ds \forall x \in \R: \, \)

\(\ds \paren {-x}^3\)

\(=\)

\(\ds \paren {-1}^3 x^3\)

\(\ds \)

\(=\)

\(\ds -x^3\)

Hence the result by definition of odd function.

$\blacksquare$

Sources

• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(h)}$ Even and Odd Functions