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Definition of an even function?
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All Flashcards in Topic 2.14
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2.14.18 cards
Definition of an even function?
f(−x) = f(x) — symmetric in the y-axis (e.g. x², cos x).
Definition of an odd function?
f(−x) = −f(x) — symmetric about the origin (e.g. x³, sin x).
How do you classify a function as odd/even?
Compute f(−x): if it equals f(x) it's even; if it equals −f(x) it's odd; otherwise neither.
Integral of an odd function over [−a, a]?
0 — the halves cancel.
Integral of an even function over [−a, a]?
2 × ∫₀ᵃ f(x) dx.
Is x³ − 4x odd, even or neither?
Odd: f(−x) = −x³ + 4x = −(x³ − 4x) = −f(x).
Which powers appear in an even polynomial?
Only even powers (and a constant); odd polynomials have only odd powers.
Which function is both odd and even?
Only f(x) = 0.
2.14.28 cards
When does a function have an inverse?
When it's one-to-one (each output comes from exactly one input — passes the horizontal-line test).
What do you do if a function isn't one-to-one?
Restrict its domain to a stretch where it IS one-to-one, then invert.
How do you find an inverse algebraically?
Write y = f(x), make x the subject, then swap x and y.
What is a self-inverse function?
One that is its own inverse: f(f(x)) = x, so f⁻¹ = f; its graph is symmetric in y = x.
Two classic self-inverse functions?
f(x) = 1/x and f(x) = a − x.
How are the domain and range of f related to f⁻¹?
The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
Largest domain for cos x to have an inverse?
[0, π] — where cos is one-to-one (the arccos domain).
Restrict x² so it has an inverse — what's f⁻¹?
On x ≥ 0, f⁻¹(x) = √x.
Topic 2.14 study notes
Full notes & explanations for Odd, even & self-inverse (HL only)
Math AA exam skills
Paper structures, command terms & tips
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