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I am given the piecewise function $f: \mathbb{Z} \to \mathbb{Z},$ $ f(x) =\left\{ \begin{matrix} 2n, & \text{if }n \text{ is even} \\ n, & \text{if }n \text{ is odd} \end{matrix}\right.$

I need to find the image of this function. So far I have tried plotting out a list of points, but I am unable to find a pattern. Is there a better approach to this problem?

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  • $\begingroup$ You have a piecewise function, so begin by focusing only on one piece at a time. For instance, what's the image of $f$ under the set of odd integers? $\endgroup$ – John Griffin Nov 9 '17 at 4:06
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    $\begingroup$ So $\text{im}(f)$ contains all odd integers, as well as all integer multiples of 4. But how do I express that as one set? Do I just say $\{n \mid n $ is odd, or $ n=4t, t \in \mathbb{Z} \}$? $\endgroup$ – jeanquilt Nov 9 '17 at 4:11
  • $\begingroup$ Exactly @jeanquilt. $\endgroup$ – TRUSKI Nov 9 '17 at 4:45
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Certainly the image contains all the odd integers. Now you want to know which even integers it contains. Notice that

$f(2) = 4 ,f(4) = 8, f(6) = 12$

How astounding ... $$f(2n) = 2(2n) = 4n$$

Thus the image contains all the odd integers and all the multiples of $4$.

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