Square Function is Even

Theorem

The square function:

$\map f x = x^2$

is an even function.

Proof

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

\(\ds \paren {-x}^2\)

\(=\)

\(\ds \paren {-x} \times \paren {-x}\)

\(\ds \)

\(=\)

\(\ds \paren {-1}^2 x^2\)

\(\ds \)

\(=\)

\(\ds x^2\)

Hence the result by definition of even 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): even function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): even function