Symmetry: Even & Odd Functions
Some functions have symmetry you can use. An even function mirrors across the y-axis; an odd function looks the same after a 180-degree turn about the origin. Many are neither.
There is a clean algebra test: substitute for and compare. If nothing changes, the function is even; if every sign flips, it is odd.
The two symmetry tests
A function is even if — replacing with leaves it unchanged. Its graph is symmetric across the y-axis, like a parabola.
A function is odd if — every term changes sign. Its graph has rotational symmetry about the origin, like .
The parabola below is even: its left and right halves are perfect mirror images across the y-axis, so and land at the same height.
A shortcut with exponents
For polynomials, even functions have only even-degree terms (and a constant is even-degree zero). Odd functions have only odd-degree terms.
So has all even powers and is even. A mix of even and odd powers, like , is neither.
Worked examples
Example 1: classify a polynomial
Is even, odd, or neither?
Answer: Even
Example 2: an odd function
Show is odd.
Answer: Odd, since
Example 3: a function that is neither
Is even, odd, or neither?
Answer: Neither
Try one yourself
Common questions
Can a function be both even and odd?
Only the zero function is both. Every other function is even, odd, or neither.
What is the fastest check for a polynomial?
Look at the exponents. All even powers (a constant counts as even) means even; all odd powers means odd; a mix means neither.
Why do these names matter?
Symmetry cuts work in half — knowing one side of the graph gives you the other — and it shows up in trig and calculus later.
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