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How do I prove this function is injective and surjective:

$h(n)=\begin{cases}f((n+1)/2),&\text{ if }n\text{ is odd}\\ g(n/2),&\text{ if }n\text{ is even}\end{cases}$

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    You can't prove that this function is injective or surjective, without knowing some details about $f$ and $g$. Can you please post the whole question?2012-09-17

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The details will depend on $f$ and $g$, but the general procedure is no different from the one that you’d use if the function were not defined by cases.

To show that $h$ is injective, show that if $h(m)=h(n)$, then $m=n$. This potentially involves considering three cases: $m$ and $n$ both odd, $m$ and $n$ both even, and one of $m$ and $n$ odd and the other even.

To show that $h$ is surjective (onto what set?), you have to show that for each $x$ in the codomain of $h$ there is an integer $n$ such that $h(n)=x$. Typically this will be done by working out what things are in the ranges of $f$ and $g$ and showing that each $x$ in the codomain of $h$ really can be ‘hit’ by one of the cases.

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    Okay, I went ahead and deleted my comments (they didn't make sense anymore in any case)2012-09-17