Key Idea: This topic tests a function's symmetry: odd, even or neither. It also restricts domains so an inverse exists, and spots functions that are their own inverse. It is mainly a Paper 1 skill.
Even: f(−x) equals f(x). Mirror in the y-axis. Only even powers and constants.
Odd: f(−x) = −f(x). Rotation about the origin. Only odd powers, no constant. Integral over [−a, a] is 0.
Inverses: domain restriction & self-inverse
- An inverse exists only if f is one-to-one (horizontal-line test).
- Not one-to-one? Restrict the domain, e.g. cos x on [0, π].
- Self-inverse: f(f(x)) = x, graph symmetric in y = x.
- The domain of f⁻¹ is the range of f.
To prove odd or even, work out f(−x) and compare it with f(x).
Is x⁴ − x² + 1 odd, even or neither? Even: changing x to −x changes nothing.
∫ from −2 to 2 of an odd function? 0.
Largest a so cos x has an inverse on 0 ≤ x ≤ a? a = π.
How do you show f is self-inverse? Apply f twice and show you get x back.