To express the function shown in the graph as a piecewise function, we need to carefully examine the different parts of the graph and the domains over which each part of the function is defined.

Analyzing the Graph:

1. Left Horizontal Line Segment:

   • This is a horizontal line at $y = 4$ extending to the left from $x = -1$ and continuing indefinitely.
   • There is an open circle at $x = -1$, which means the value at $x = -1$ is not included in this part of the function.
   • Domain: $x < -1$

2. Right Diagonal Line Segment:

   • This is a line with a positive slope starting from $x = 1$ at $y = 5$ and increasing.
   • There is an open circle at $x = 1$, so the value at $x = 1$ is not included.
   • From the graph, we can calculate the slope between the points $(1, 5)$ and $(6, 9)$.
     • Slope $m = \frac{9 - 5}{6 - 1} = \frac{4}{5}$
     • Equation of the line: $y - 5 = \frac{4}{5}(x - 1)$, simplifying to $y = \frac{4}{5}x + \frac{1}{5}$.
   • Domain: $x > 1$

Piecewise Function:

The function can be expressed as: