How to find image of a function

Question

Given a function $f: \mathbf{N}_0 \to \mathbf{N}_0$, defined $$ f(x) = \begin{cases} x+3 & \text{if } x \in \mathbf{N}_{\text{even}} \\ x-1 & \text{if } x \in \mathbf{N}_{\text{odd}} \end{cases} $$

How can I find the image $f( $$\mathbf{N}_{\text{even}}$ )?

Answer 1

Note: Every even number $n$ can be written as

$n=2k, k=0,1,2 ......$.

Hence:

$f(2k) = 2k+3, k=0,1,2,.....$.

Finally:

$f(\mathbb{N_{even}}) = $

{$2k+3| k \in \mathbb{N}$} $= ${$3,5,7,........$}.

Answer 2

Well, $\ f(\mathbb{N}_{even})\ $ is simply just $\ f|_{\mathbb{N}_{even}}:\mathbb{N}_{even} \rightarrow \mathbb{N}_{odd}\ $ defined by $\ f(x) = x+3\ $. So it follows that, $\ f(\mathbb{N}_{even})=\{ 0+3, 2+3, 4+3, \dots \} \ $ so then the image is simply $\ \{3,5,7,\dots \}$.