Hello, Stuckasaurus!
$\displaystyle f(x) \;=\;\begin{Bmatrix}\sin\left(\frac{\pi}{2}x\right ) & [0,2] \\ \\[4mm] 0 & [2,4] \\ f(x+4) & {\color{red}?}\end{Bmatrix}$
I'm not sure what that third entry means.
It probably means that this graph is repeated over and over,
. . but I'll graph just one "cycle".
The brackets indicate intervals.
. . $\displaystyle [0,2]$ means: .$\displaystyle 0 \leq x \leq 2$
Brackets mean the endpoints are included.
If the endpoints are not included, parentheses are used.
. . $\displaystyle (1,3)$ means: .$\displaystyle 1 < x < 3$
And, yes, they can be "mixed".
. . $\displaystyle (0, 4]$ means: .$\displaystyle 0 < x \leq 4$
This problem says:
If $\displaystyle 0 \leq x \leq 2$, the graph is a sine function: .$\displaystyle y \:=\:\sin\left(\frac{\pi}{2}x\right)$
If $\displaystyle 2 \leq x \leq 4$, the graph is $\displaystyle y \:=\:0$ ... the xaxis.
The graph looks like this: Code:

1+ *
 * *
 * *
* *

  *        * * * * * * * * *  
 2 4
$\displaystyle f(x) \;=\;\begin{Bmatrix}0 & 1 < x < 0 \\ \\[4mm]
\frac{1}{2} & x \in Z \\ \\[4mm] 1 & 0 < x < 1 \\ f(x+2) & {\color{red}?}\end{Bmatrix} $
This function says:
If $\displaystyle x$ is between 1 and 0, the graph is $\displaystyle y = 0$, the xaxis.
If $\displaystyle x$ is an integer, the graph is ½.
If $\displaystyle x$ is between 0 and 1, the graph is $\displaystyle y = 1$
And the graph looks like this: Code:

1o=======o

*

  o=======o    +  
1  1