AHL 2.14 (HL)—Odd/even and inverse functions

Use symmetry and domain restrictions to analyse inverses.

An even function satisfies f(−x)=f(x), an odd function satisfies f(−x)=−f(x). An inverse exists as a function only after the original is one-to-one on its chosen domain.

Worked example

f(x)=x² is even but not one-to-one on ℝ. Restricting to x≥0 gives f⁻¹(x)=√x; the graph reflects across y=x.

Check the domain before finding an inverse; the same formula can produce different inverse branches.

Symmetry does not imply invertibility: even functions usually map two inputs to one output.

Periodic and self-inverse examples: $\cos x$ is even and $2\pi$-periodic, while $\sin x$ is odd and $2\pi$-periodic. The function $f(x)=1/x$ on $x\ne0$ is odd and self-inverse because $f(f(x))=1/(1/x)=x$. Self-inverse means $f^{-1}=f$; it does not mean every input is fixed by $f$.