Onto Functions — Cheat sheet

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Key method

For $f:A\to B$ , surjectivity means: for every $y\in B$ , there exists $x\in A$ with $f(x)=y$ . Verify both that the constructed input belongs to $A$ and that substitution returns $y$ .

Example

Take $f:\mathbb R\to\mathbb R$ , $f(x)=3x-2$ . Given any real $y$ , set $x=(y+2)/3$ . This is real and $f(x)=3(y+2)/3-2=y$ . Therefore the function is onto. On domain $[0,\infty)$ with the same codomain, it is not onto: its range starts at $-2$ .

Avoid this mistake

Checking a few targets is evidence, not a proof for every target. Use a symbolic arbitrary $y$ .