Question

what are the domain and range of this relation?
image of a coordinate plane with points plotted
options:

1. domain: {-3, -2, 1, 2, 3}; range: {-3, 1, 4, 5}
2. domain: {-3, -2, 1, 2, 3}; range: {-2, 0, 1, 3, 4}
3. domain: {-2, 0, 2, 3}; range: {-3, 1, 4, 5}
4. domain: {-2, 0, 2, 3}; range: {-2, 0, 1, 3, 4}

Explanation:

Step1: Identify x - coordinates (domain)

The points on the graph have x - coordinates: - 2, 0, 2, 3 (and also check the leftmost point at x=-5? Wait, no, looking at the plotted points: the points are at (-5, 0)? Wait, no, re - examining the graph: the points are: (-2,5), (-2,1), (0,4), (2,1), (3, - 3), and (-5,0)? Wait, no, the leftmost point is at x=-5, y = 0? Wait, no, maybe I misread. Wait, the x - coordinates of the plotted points: let's list the points by their (x,y) coordinates. From the graph:

• Point 1: (-5, 0)
• Point 2: (-2, 5)
• Point 3: (-2, 1)
• Point 4: (0, 4)
• Point 5: (2, 1)
• Point 6: (3, - 3)

Wait, no, maybe the options are different. Wait, the options for domain: let's check the options. The options have domain as {-3, - 2,1,2,3} or {-2,0,2,3}. Wait, maybe I misread the x - coordinates. Let's look again. The x - axis labels: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. The plotted points:

• At x=-2: two points (y = 5 and y = 1)
• At x = 0: one point (y = 4)
• At x = 2: one point (y = 1)
• At x = 3: one point (y=-3)
• And is there a point at x=-5? Wait, the leftmost point is at x=-5, y = 0? But the options don't have -5. Wait, maybe the leftmost point is a mistake, or maybe I misread. Wait, the options for domain: the first two options have domain {-3, - 2,1,2,3}, the last two have {-2,0,2,3}. Let's check the y - coordinates (range). The y - values of the points: 5,1,4,1, - 3. So unique y - values: - 3,1,4,5. Now check the domain (x - values): x=-2, x = 0, x = 2, x = 3? Wait, no, maybe the point at x=-2 (two points), x = 0, x = 2, x = 3. Wait, the third option has domain {-2,0,2,3} and range {-3,1,4,5}, which matches the y - values (-3,1,4,5) and x - values (-2,0,2,3). Let's verify:
• Domain is the set of all x - values of the relation. The x - values of the points are - 2, 0, 2, 3 (and maybe -5? But the options don't have -5, so maybe the leftmost point is not part of the relation in the options' context).
• Range is the set of all y - values. The y - values are 5,1,4, - 3. So range is {-3,1,4,5}. And domain is {-2,0,2,3}. So the correct option is the third one: domain: {-2,0,2,3}, range: {-3,1,4,5}

Step2: Match with options

Now, looking at the options:

• First option: domain {-3, - 2,1,2,3}, range {-3,1,4,5} → x=-3 is not a point, so wrong.
• Second option: domain {-3, - 2,1,2,3}, range {-2,0,1,3,4} → wrong range and wrong domain.
• Third option: domain {-2,0,2,3}, range {-3,1,4,5} → matches x - values (-2,0,2,3) and y - values (-3,1,4,5)
• Fourth option: domain {-2,0,2,3}, range {-2,0,1,3,4} → wrong range.

Answer:

domain: \{-2, 0, 2, 3\}\\
range: \{-3, 1, 4, 5\} (corresponding to the third option in the given choices, i.e., the option with domain \{-2, 0, 2, 3\} and range \{-3, 1, 4, 5\})