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}}$ )?
Best Answer
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 $\ \{3,5,7,\dots \}$.