To determine the piecewise function $f(x)$ from the graph, we need to analyze the two different line segments and the corresponding intervals they represent.

Step 1: Analyzing the first line (from left to $x = 3$)

• This line starts from the left side (negative $x$-values) and extends until $x = 3$, where there is a closed circle at the point $(3, 3)$.

• This means the first piece of the function is valid for $x \leq 3$.

• The slope of this line can be calculated between two points, for example, $(-9, -9)$ and $(0, 0)$.

  $\text{slope} = \frac{(0 - (-9))}{(0 - (-9))} = 1$ Therefore, the equation of this line is:

  $y = x$ So, $f(x) = x$ for $x \leq 3$.

Step 2: Analyzing the second line (from $x = 3$ to $x = 9$)

• The second line starts from $x = 3$ with an open circle at the point $(3, 3)$, meaning the function is not defined at $x = 3$, but it continues for $x > 3$.

• The slope of this line can be calculated between $(3, 3)$ and $(9, 0)$:

  $\text{slope} = \frac{0 - 3}{9 - 3} = \frac{-3}{6} = -\frac{1}{2}$ Using the point-slope form of a line:

  $y - 3 = -\frac{1}{2}(x - 3)$ Simplifying:

  $y = -\frac{1}{2}x + \frac{9}{2}$ So, $f(x) = -\frac{1}{2}x + \frac{9}{2}$ for $x > 3$.

Step 3: Writing the final piecewise function