Why One-to-One and Onto Matter — Cheat sheet

Key method

If $f:A\to B$ is bijective, then $f^{-1}:B\to A$ satisfies

Surjectivity guarantees existence of the reverse assignment; injectivity guarantees uniqueness.

Example

The function $f:[0,\infty)\to[0,\infty)$, $f(x)=x^2$, is bijective. Its inverse is $\sqrt x$. If its domain expands to all real numbers, uniqueness is lost. If its codomain expands to all real numbers, some targets have no preimage.

Avoid this mistake

An inverse is not a reciprocal: $f^{-1}(x)$ usually differs from $1/f(x)$.