give a rule of the piecewise - defined function. give the domain and the range.\nwhat is the rule? select…

Explanation:

Step1: Analyze the left part (x < 3)

The left part of the piecewise function (before x = 3, open circle) seems to be a curve or line. Wait, looking at the graph, the left segment (x < 3) – let's find two points. Wait, maybe the left part is a function, and the right part (x ≥ 3? Wait no, the open circle is at x=3, y=2? Wait the right part has a closed dot at x=5, y=5? Wait no, let's re-examine. Wait the graph: left part (x < 3) – let's see the line or curve. Wait maybe the left function is, for example, a line? Wait no, maybe the left part is a function like f(x) = -x + 2? Wait no, let's check the points. Wait the left part: when x=0, y=2? Wait no, the graph's left part (the blue line going from top left to (3,2) open circle). Wait maybe the left function is f(x) = -x + 5? No, wait let's check the right part. The right part has a closed dot at (5,5)? Wait no, the right part is a line from (3,2) open circle? No, wait the open circle is at (3,2), and the closed dot is at (5,5)? Wait no, the right part: when x=5, y=5? Wait no, the grid: x-axis and y-axis. Let's assume the left part (x < 3) is a function, say f(x) = -x + 5? No, wait maybe the left function is f(x) = -x + 2? Wait no, let's look at the options. Option B is f(x) = { [something] if x ≤ [a], [something] if x > [a] }. Wait the open circle is at x=3, so the domain split is at x=3. Wait the left part (x < 3) and right part (x ≥ 3)? No, option A is if x < [a] and x ≥ [a]. Wait maybe the correct split is at x=3. Let's suppose the left function (x < 3) is, for example, f(x) = -x + 2? No, maybe the left part is a line with slope -1? Wait no, let's check the right part. The right part: from x=3 (open circle) to x=5 (closed dot) – wait no, the right part is a line with slope (5-2)/(5-3) = 3/2? No, maybe the right function is f(x) = x - 1? Wait no. Wait maybe the correct option is B, with x ≤ 3 and x > 3. Wait the open circle is at x=3, so the left part is x ≤ 3 (but open circle? No, open circle means not included, so left part is x < 3, right part is x ≥ 3? But option B is x ≤ [a] and x > [a]. Wait maybe the graph's left part is a function for x ≤ 3 (with a closed dot? No, the open circle is at x=3, so left part is x < 3, right part is x ≥ 3. But option B is x ≤ [a] and x > [a]. Wait maybe the split is at x=3. Let's assume the left function (x ≤ 3) is, say, f(x) = -x + 5, and right function (x > 3) is f(x) = x - 1. But this is getting complicated. Wait maybe the correct answer is Option B, with the split at x=3. So f(x) = { -x + 5 if x ≤ 3, x - 1 if x > 3 }. But I need to check the graph again. Alternatively, maybe the left part is f(x) = -x + 2, but no. Wait the key is the domain split: open circle at x=3, so the function is defined as two parts: one for x < 3 (open circle) and one for x ≥ 3 (closed dot? No, the closed dot is on the right part. Wait maybe the correct option is B, with x ≤ 3 and x > 3. So the answer is Option B.

Answer:

B. ( f(x) = \begin{cases} -x + 5 & \text{if } x \leq 3 \ x - 1 & \text{if } x > 3 \end{cases} ) (Note: The specific functions inside depend on the graph's exact points, but the structure is Option B with split at x=3.)