Replace x with −x: Even: f(−x) = f(x) — mirror symmetry in the y-axis (like x², cos x).
Odd: f(−x) = −f(x) — 180° rotational symmetry about the origin (like x³, sin x).
To classify, work out f(−x) and compare with f(x).
IB-style question — show a function is odd
Show that f(x) = x³ − 4x is an odd function.
Step by step
- Replace x with −x.
- Factor out −1 and compare with f(x).
Final answer
f(−x) = −f(x), so f is odd.
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Odd over a symmetric interval = 0: Over an interval [−a, a]:
odd function → the integral is 0 (the two halves cancel).
even function → the integral is 2 × the half from 0 to a.
IB-style question — use the symmetry
Without finding an antiderivative, evaluate ∫₋₂² (x³ − 4x) dx.
Step by step
- x³ − 4x is odd (shown above), and the interval [−2, 2] is symmetric.
- An odd function over a symmetric interval integrates to 0.
Final answer
0 — the positive and negative halves cancel.