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$f:Z \rightarrow Z$

$f(x) = x-5$ , when $x$ is odd

$f(x) = x+3$ , when $x$ is even

It seems that it is not onto, because not all integers are covered, but how do you show this?

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    But it is onto! look at the individual y-sets for each of those functions. You see that the first function produceds a y-set of all even numbers, and the second function produces a y-set of all odd numbers. Combining them makes all the integers.2012-11-09
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    Following on mathguy's comment: if X is odd, what is $f^{-1}(X)?$ and if X is even, what is $f^{-1}(X)$>2012-11-09

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