Even and odd functions

Recognise even functions (symmetric in the y-axis) and odd functions (symmetric about the origin), and test any function algebraically by substituting −x.

Two kinds of symmetry

Even: $f(-x) = f(x)$ for all $x$ in the domain.

The graph has line symmetry in the $y$-axis: reflecting it in the $y$-axis leaves it unchanged.

Examples: $x^2$, $x^4 - 3x^2$, $\dfrac{1}{x^2}$.

Odd: $f(-x) = -f(x)$ for all $x$ in the domain.

The graph has point symmetry about the origin: a half-turn about $O$ leaves it unchanged.

Examples: $x^3$, $x^5 - 2x$, $\dfrac{1}{x}$.

Even powers, odd powers

A polynomial is even if it has only even powers of x (a constant counts as even), and odd if it has only odd powers.

Try this

Set $b$ and $d$ to zero. Check the graph is symmetric in the $y$-axis.
Now add a constant term in your head: $x^3 + 1$. Is it still odd?

Testing algebraically

Example 1●●

Test $f(x) = \dfrac{x}{x^2 + 1}$ for evenness and oddness.

$f(-x) = \dfrac{-x}{(-x)^2 + 1} = \dfrac{-x}{x^2 + 1}$

Replace every x with (−x).

$= -f(x)$, so $f$ is odd.

Example 2●●

Is $g(x) = x^2 + x$ even, odd or neither?

$g(-x) = x^2 - x$

This is neither $g(x)$ nor $-g(x)$, so $g$ is neither even nor odd.

One counterexample is enough: g(1) = 2 but g(−1) = 0.

Flashcards

Even function

$f(-x) = f(x)$. Line symmetry in the $y$-axis.

Odd function

$f(-x) = -f(x)$. Point symmetry about the origin.

Polynomial test

Only even powers: even. Only odd powers: odd.

Odd function defined at 0

Must pass through the origin.

Practice

$f(x) = x^4 - 5x^2 + 1$ is:
$f(x) = 2x^3 - x$ is:
$f(x) = x^3 + 1$ is:
$f$ is odd and $f(2) = 6$. Find $f(-2)$.
$g$ is even and $g(-3) = 9$. Find $g(3)$.
Which graph has point symmetry about the origin?
$f(x) = \dfrac{1}{x^2 - 4}$ is:
An odd function is defined at $x = 0$. What must be true?