Functions That Are Not Onto — Cheat sheet

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Key method

To disprove onto behavior, exhibit a single element of the codomain with no preimage. You must know the codomain before making this judgment.

Example

For $f:\mathbb R\to\mathbb R$ with $f(x)=x^2$ , the target $-1$ cannot be reached by a real input. Thus $f$ is not onto. If instead the declared codomain is $[0,\infty)$ , every target $y$ has preimage $\sqrt y$ , so the same formula and domain become onto that codomain.

Avoid this mistake

A function can be one-to-one without being onto, or onto without being one-to-one. These are independent properties.