Even and Odd Functions Math Example 3

Follow the full solution, then compare it with the other examples linked below.

medium

h(x) = x^3 + x^2

even, odd, or neither?

1
$h(-x) = (-x)^3 + (-x)^2 = -x^3 + x^2$ . Compare: $h(x) = x^3 + x^2$ and $-h(x) = -x^3 - x^2$ .
2
$h(-x) \neq h(x)$ and $h(-x) \neq -h(x)$ , so $h$ is neither even nor odd.

Answer

\text{Neither even nor odd}

A function that mixes even and odd powers of

x

is typically neither even nor odd. Any function can be decomposed into even and odd parts:

f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}

. Here the even part is

x^2

and the odd part is

x^3

About Even and Odd Functions

An even function satisfies $f(-x) = f(x)$ (symmetric about $y$ -axis); an odd function satisfies $f(-x) = -f(x)$ (rotational symmetry about origin).

Determine whether [formula] is even, odd, or neither.

Determine whether [formula] is even, odd, or neither.

Prove that the product of two odd functions is an even function.