One-to-one but not onto

Onto but not one-to-one

Neither one-to-one nor onto

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Solution

The floor function $f(x) = [x]$ maps every real number to the greatest integer less than or equal to $x$. Different real numbers can have the same floor value, so it is not one-to-one, but for every integer $n \in \mathbb{Z}$, there exists an $x \in \mathbb{R}$ such that $[x]=n$. Hence, it is onto but not one-to-one.