The graph shows three distinct segments that need to be expressed as a piecewise function:

1. For $x < -2$:
   The graph is a horizontal line with a constant value at $y = -2$.

2. For $-2 \leq x \leq 0$:
   The graph is another horizontal line at $y = 0$. However, there is an open circle at $x = 0$, indicating that the value is not included at that point. The notation should reflect this discontinuity at $x = 0$.

3. For $x > 0$:
   The graph is a line with positive slope, passing through the point $(0, 1)$ and extending upwards. To find the equation of this line, we can use the points $(0, 1)$ and $(2, 2)$ to determine the slope: $\text{slope} = \frac{2 - 1}{2 - 0} = \frac{1}{2}.$ Using the point-slope form of a line, the equation is: $y - 1 = \frac{1}{2}(x - 0) \quad \Rightarrow \quad y = \frac{1}{2}x + 1.$

Now, let's write the piecewise function: