1. Definitions: Even and Odd Functions
A function $f$ is called even if, for every $x$ in its domain,
$$ f(-x) = f(x). $$In words: replacing the input with its negative does not change the output.
A function $f$ is called odd if, for every $x$ in its domain,
$$ f(-x) = -f(x). $$In words: replacing the input with its negative negates the output.
Critical Prerequisite — Domain Symmetry
Before you can even ask whether a function is even or odd, its domain must be symmetric about the origin: $x$ is in the domain if and only if $-x$ is in the domain.
Example of failure: Consider $f(x) = x^2$ defined only on $[0, \infty)$. The formula satisfies $f(-x) = f(x)$ wherever both sides make sense, but the domain is not symmetric (e.g., $x = 3$ is in the domain but $x = -3$ is not). Therefore $f$ cannot be classified as even in the standard sense.
2. Geometric Meaning
Even Functions — y-axis Symmetry
The graph of an even function is symmetric about the $y$-axis. If you fold the graph along the $y$-axis, the left half lands exactly on the right half.
Visually: for every point $(a, b)$ on the graph, the point $(-a, b)$ is also on the graph.
Odd Functions — Origin Symmetry
The graph of an odd function is symmetric about the origin. It has 180° rotational symmetry: rotating the graph half a turn around $(0,0)$ leaves it unchanged.
Visually: for every point $(a, b)$ on the graph, the point $(-a, -b)$ is also on the graph.
Even: $f(x) = x^2$ (y-axis symmetry)
Odd: $f(x) = x^3$ (origin symmetry)
- If the graph looks the same when you reflect it across the $y$-axis → the function is even.
- If the graph looks the same when you rotate it 180° around the origin → the function is odd.
- If neither symmetry holds → the function is neither.