Even, Odd & Symmetry

You’ll be able to

Test a function algebraically for even or odd symmetry
Connect even symmetry to reflection across the y-axis and odd symmetry to rotation about the origin
Predict the parity of a sum or product of functions of known parity

Symmetry is a statement about f(−x)

A function is even if f(−x) = f(x) for every x in its domain, and odd if f(−x) = −f(x). Even means the graph is unchanged by reflection across the y-axis; odd means it is unchanged by a 180° rotation about the origin. Most functions are neither — and "neither" is a legitimate, common answer, not a sign that you made an arithmetic error.

Parity tests

even: f(−x) = f(x) · odd: f(−x) = −f(x) · neither: some x where both fail

The names come from powers: xⁿ is even when n is even and odd when n is odd. Testing means substituting −x and simplifying, then comparing to f(x) and to −f(x).

Worked example

Classify f(x) = x⁴ − 3x², g(x) = x³ − x, and h(x) = x² + x.

1.f(−x) = (−x)⁴ − 3(−x)² = x⁴ − 3x² = f(x). Even.
2.g(−x) = (−x)³ − (−x) = −x³ + x = −(x³ − x) = −g(x). Odd.
3.h(−x) = (−x)² + (−x) = x² − x. Compare: h(x) = x² + x, so h(−x) ≠ h(x). And −h(x) = −x² − x, so h(−x) ≠ −h(x) either.

Answer: f is even, g is odd, and h is neither. Note the pattern: f has only even powers, g has only odd powers, and h mixes them — which is precisely why h has no symmetry.

The polynomial shortcut, and its limits

For a polynomial the test is immediate: only even powers means even, only odd powers means odd, a mix means neither. A constant term counts as an even power, x⁰, so x³ + 1 is neither despite looking almost odd. This shortcut does not extend past polynomials — for √x, |x|, eˣ, sin x or a rational function you must actually substitute −x. (For the record: |x| is even, sin x is odd, cos x is even, and eˣ is neither.)

Watch out

A single test value proves nothing. If f(−2) = f(2), the function might be even — or you might have landed on a coincidence. Even and odd are claims about every x, so the argument has to be algebraic.

Checkpoint

Classify f(x) = (x³ + x)/(x² + 1).

How parity combines

Parity behaves like the arithmetic of even and odd integers, which is why the names were chosen. even × even = even. odd × odd = even. even × odd = odd. Sums are different and less forgiving: even + even = even and odd + odd = odd, but even + odd is neither — you cannot mix. And exactly one function is both even and odd: f(x) = 0, since it is the only function equal to its own negative.

Combining parities

product: even·even = even, odd·odd = even, even·odd = odd · sum: like + like keeps the type, mixed gives neither

Composition too: f∘g is even whenever g is even, whatever f is — because g(−x) = g(x) means the composition never sees the sign.

Checkpoint

If f is odd and g is even, what can you say about the product p(x) = f(x)·g(x)?

On the exam

Symmetry halves your work. If you know a function is even, analyzing x ≥ 0 determines the whole graph by reflection. Free-response questions sometimes hand you symmetry precisely so you can transfer a computed value from one side to the other.

Answer the 2 checkpoints as you read.