Question

what is the range of this relation?
options:
{-5, -3, -1, 3}
{-5, -1, 3}
{-5, -3, 3, 4}
{-5, -1, 3, 4}

Explanation:

Step1: Identify the points

First, we find the coordinates of the green points on the graph. Let's list their coordinates:
- The point at \( x = -4, y = -5 \) (wait, no, looking at the grid, let's check each green dot:
Wait, let's re-examine the graph. The green dots are at:
- \( (-4, -5) \)? Wait, no, the x-axis and y-axis: Let's see the positions. Wait, the first green dot (bottom left) is at \( x = -4, y = -5 \)? Wait, no, the grid lines: each square is 1 unit. Let's check the y-values (range is the set of y-values of the relation).

Wait, the green points:
1. At \( x = -4 \), \( y = -5 \)
2. At \( x = -3 \), \( y = -1 \)
3. At \( x = -1 \), \( y = 3 \)
4. At \( x = 3 \), \( y = -1 \)
5. At \( x = 4 \), \( y = -5 \)

Wait, no, maybe I misread. Wait, the problem is about the range, which is the set of all y-values (outputs) of the relation. So we need to collect all unique y-values from the points.

Wait, let's list the y-coordinates of each green point:

- The point at \( x = -4 \): y = -5
- The point at \( x = -3 \): y = -1
- The point at \( x = -1 \): y = 3
- The point at \( x = 3 \): y = -1
- The point at \( x = 4 \): y = -5

Now, collect the unique y-values: -5, -1, 3. Wait, no, wait: Wait, maybe I made a mistake. Wait, let's check again. Wait, the options are:

First option: {-5, -3, -1, 3}

Second: {-5, -1, 3}

Third: {-5, -3, 3, 4}

Fourth: {-5, -1, 3, 4}

Wait, maybe I misread the x and y. Wait, the range is the set of y-values. Let's check each green dot's y-coordinate:

1. The bottom-left green dot: y = -5 (what's its x? x = -4? But the options have -5, -3, etc. Wait, maybe the points are:

Wait, the first green dot (bottom left) is at (x=-4, y=-5)? No, maybe the x is -4, but the y is -5. Then the next green dot is at (x=-3, y=-1). Then at (x=-1, y=3). Then at (x=3, y=-1). Then at (x=4, y=-5). So the y-values are -5, -1, 3. Wait, but the first option is {-5, -3, -1, 3}, but -3 is not a y-value. Wait, maybe I messed up the coordinates.

Wait, maybe the points are:

- (x=-4, y=-5)
- (x=-3, y=-1)
- (x=-1, y=3)
- (x=3, y=-1)
- (x=4, y=-5)

So the y-values are -5, -1, 3. Wait, but the first option is {-5, -3, -1, 3}, but -3 is not a y-value. Wait, maybe I misread the points. Wait, let's look at the options. The options are:

1. {-5, -3, -1, 3}

2. {-5, -1, 3}

3. {-5, -3, 3, 4}

4. {-5, -1, 3, 4}

Wait, maybe the points are:

- (x=-4, y=-5)

- (x=-3, y=-1)

- (x=-1, y=3)

- (x=3, y=-1)

- (x=4, y=-5)

So the y-values are -5, -1, 3. But wait, the first option has -3, which is not a y-value. Wait, maybe I made a mistake. Wait, maybe the points are:

Wait, the green dot at x=-3: y=-1? No, maybe the green dot at x=-3: y=-1, x=-1: y=3, x=-4: y=-5, x=3: y=-1, x=4: y=-5. So the unique y-values are -5, -1, 3. But the first option is {-5, -3, -1, 3}, which includes -3. Wait, maybe I misread the y-coordinate of the x=-3 point. Wait, maybe the x=-3 point has y=-1, x=-1 has y=3, x=-4 has y=-5, x=3 has y=-1, x=4 has y=-5. So the y-values are -5, -1, 3. But the first option is {-5, -3, -1, 3}, which is wrong. Wait, maybe the points are:

Wait, another approach: Range is the set of all y-values (outputs) of the relation. So we need to list all the y-coordinates of the ordered pairs in the relation.

Looking at the graph, the green dots are at:

1. (-4, -5)

2. (-3, -1)

3. (-1, 3)

4. (3, -1)

5. (4, -5)

Now, collect the y-values: -5, -1, 3. Wait, but -5 appears twice, -1 appears twice, 3 appears once. So the unique y-values are -5, -1, 3. But the first option is {-5, -3, -1, 3}, which includes -3. Wait, maybe I misread the x=-3 point. Wait, maybe the x=-3 point has y=-1, x=-1 has y=3, x=-4 has y=-5, x=3 has y=-1, x=4 has y=-5. So the y-values are -5, -1, 3. But the first option is {-5, -3, -1, 3}, which is incorrect. Wait, maybe the points are:

Wait, maybe the green dot at x=-3 has y=-1, x=-1 has y=3, x=-4 has y=-5, x=3 has y=-1, x=4 has y=-5. So the y-values are -5, -1, 3. But the first option is {-5, -3, -1, 3}, which is wrong. Wait, maybe I made a mistake. Wait, the options:

First option: {-5, -3, -1, 3}

Second: {-5, -1, 3}

Third: {-5, -3, 3, 4}

Fourth: {-5, -1, 3, 4}

Wait, maybe the points are:

- (x=-4, y=-5)

- (x=-3, y=-1)

- (x=-1, y=3)

- (x=3, y=-1)

- (x=4, y=-5)

So the y-values are -5, -1, 3. But the first option includes -3, which is not a y-value. Wait, maybe the x=-3 point has y=-1, x=-1 has y=3, x=-4 has y=-5, x=3 has y=-1, x=4 has y=-5. So the unique y-values are -5, -1, 3. So the correct set is {-5, -1, 3}, which is the second option. Wait, but let's check again. Wait, maybe I misread the y-coordinate of the x=-3 point. Wait, the x=-3 point: looking at the graph, the y-axis is vertical. The x=-3 is at the third grid line from the left (since x=-5, -4, -3, -2, -1, 0, etc.). The y-coordinate for x=-3: the green dot is at y=-1? Wait, no, maybe the x=-3 point is at y=-1, x=-1 is at y=3, x=-4 is at y=-5, x=3 is at y=-1, x=4 is at y=-5. So the y-values are -5, -1, 3. So the range is {-5, -1, 3}, which is the second option.

Wait, but the first option is {-5, -3, -1, 3}, which includes -3. But -3 is an x-value, not a y-value. So that's a common mistake: confusing domain (x-values) and range (y-values). So the range is the set of y-values, so we need to collect all unique y-coordinates.

So the y-coordinates are -5 (from x=-4 and x=4), -1 (from x=-3 and x=3), and 3 (from x=-1). So the unique y-values are -5, -1, 3. Therefore, the range is {-5, -1, 3}.

Step2: Match with options

Now, we look at the options:

- First option: {-5, -3, -1, 3} (includes -3, which is an x-value, not y)

- Second option: {-5, -1, 3} (matches our unique y-values)

- Third option: {-5, -3, 3, 4} (includes x-values -3, 4 and y-values? No, 4 is not a y-value)

- Fourth option: {-5, -1, 3, 4} (includes 4, which is not a y-value)

So the correct range is {-5, -1, 3}.

Answer:

{-5, -1, 3} (the second option: {-5, -1, 3})