Question

what is the range of this function?

the image shows a coordinate grid with several purple points plotted. the points are at (-5, -4), (-4, 5), (-3, 0), (-2, -5), and (5, 1).

options:
{ -5, -4, -3, 5 }
{ -5, -4, 0, 1, 5 }
{ -5, -4, 1, 5 }
{ -5, -4, -3, 1, 5 }

Explanation:

Step1: Recall the definition of range

The range of a function is the set of all possible \( y \)-values (outputs) of the function. So we need to find the \( y \)-coordinates of all the plotted points.

Step2: Identify the \( y \)-coordinates of each point

- The point at \( x = -5 \): Let's look at its \( y \)-coordinate. From the graph, when \( x=-5 \), \( y = -4 \)? Wait, no, wait. Wait, let's list each point:
- Point 1: Let's see the coordinates. Let's check each purple dot:
- First, the dot at \( x=-5 \): What's its \( y \)-value? Looking at the grid, when \( x = -5 \), the \( y \)-coordinate is \( -4 \)? Wait, no, maybe I misread. Wait, let's list all the points:
- The dot at \( x=-5 \): Let's check the \( y \)-axis. Wait, the vertical axis is \( y \). So each point is \( (x, y) \). Let's find each \( (x, y) \):
- Point 1: \( (-5, -4) \) (since at \( x=-5 \), \( y=-4 \))
- Point 2: \( (-4, 5) \) (at \( x=-4 \), \( y = 5 \))
- Point 3: \( (-3, 0) \)? Wait, no, the dot at \( x=-3 \): looking at the graph, \( x=-3 \), \( y = 0 \)? Wait, no, the options have different values. Wait, maybe I made a mistake. Wait, let's check the options. Wait, the options are sets of \( y \)-values. Let's re-express:

Wait, let's list all the \( y \)-coordinates of the plotted points:

- The dot at \( x=-5 \): \( y=-4 \)
- The dot at \( x=-4 \): \( y = 5 \)
- The dot at \( x=-3 \): \( y = 0 \)? No, wait the options don't have 0 in some. Wait, no, maybe I misread the points. Wait, the other points:

Wait, there's a dot at \( x=-2 \): \( y=-5 \)

Wait, no, the options are:

Option 1: \(\{-5, -4, -3, 5\}\)

Option 2: \(\{-5, -4, 0, 1, 5\}\)

Option 3: \(\{-5, -4, 1, 5\}\)

Option 4: \(\{-5, -4, -3, 1, 5\}\)

Wait, let's list all the \( y \)-coordinates of the purple dots:

1. The dot at \( x=-5 \): \( y=-4 \)
2. The dot at \( x=-4 \): \( y = 5 \)
3. The dot at \( x=-3 \): \( y = 0 \)? No, wait the dot at \( x=-3 \) is on the \( x \)-axis, so \( y=0 \)? But the options don't have 0 in some. Wait, maybe I messed up. Wait, the dot at \( x=-2 \): \( y=-5 \)

Wait, the dot at \( x=5 \): \( y = 1 \)

Wait, let's list all the \( (x, y) \) points:

- \( (-5, -4) \)
- \( (-4, 5) \)
- \( (-3, 0) \)
- \( (-2, -5) \)
- \( (5, 1) \)

So the \( y \)-coordinates are \( -5, -4, 0, 1, 5 \)? Wait, no, \( (-2, -5) \) has \( y=-5 \), \( (-5, -4) \) has \( y=-4 \), \( (-3, 0) \) has \( y=0 \), \( (5, 1) \) has \( y=1 \), and \( (-4, 5) \) has \( y=5 \). Wait, but the options:

Wait, the second option is \(\{-5, -4, 0, 1, 5\}\). Let's check:

- \( y=-5 \) (from \( (-2, -5) \))
- \( y=-4 \) (from \( (-5, -4) \))
- \( y=0 \) (from \( (-3, 0) \))
- \( y=1 \) (from \( (5, 1) \))
- \( y=5 \) (from \( (-4, 5) \))

Yes, that matches the second option? Wait, no, wait the options:

Wait, the options are:

1. \(\{-5, -4, -3, 5\}\) – no, because we have \( 0, 1 \)
2. \(\{-5, -4, 0, 1, 5\}\) – yes, because the \( y \)-coordinates are \( -5, -4, 0, 1, 5 \)
3. \(\{-5, -4, 1, 5\}\) – no, missing \( 0 \)
4. \(\{-5, -4, -3, 1, 5\}\) – no, missing \( 0 \), and \( -3 \) is not a \( y \)-coordinate (the dot at \( x=-3 \) has \( y=0 \), not \( -3 \))

Wait, maybe I made a mistake in the \( x=-3 \) point. Let's check the graph again. The dot at \( x=-3 \) is on the \( x \)-axis, so \( y=0 \). The dot at \( x=-2 \) is at \( y=-5 \). The dot at \( x=-5 \) is at \( y=-4 \). The dot at \( x=-4 \) is at \( y=5 \). The dot at \( x=5 \) is at \( y=1 \). So the \( y \)-values are \( -5, -4, 0, 1, 5 \), which is the second option. Wait, but let's confirm:

Wait, the options are:

First option: \(\{-5, -4, -3, 5\}\) – no, because \( -3 \) is not a \( y \)-value (the \( x=-3 \) has \( y=0 \))

Second option: \(\{-5, -4, 0, 1, 5\}\) – yes, because the \( y \)-coordinates are \( -5 \) (from \( x=-2 \)), \( -4 \) (from \( x=-5 \)), \( 0 \) (from \( x=-3 \)), \( 1 \) (from \( x=5 \)), \( 5 \) (from \( x=-4 \))

Third option: \(\{-5, -4, 1, 5\}\) – no, missing \( 0 \)

Fourth option: \(\{-5, -4, -3, 1, 5\}\) – no, missing \( 0 \), and \( -3 \) is not a \( y \)-value

So the correct range is the set of \( y \)-coordinates, which are \( -5, -4, 0, 1, 5 \), so the second option.

Wait, but let me check again. Maybe I misread the points. Let's list all the purple dots:

1. At \( x=-5 \): \( y=-4 \) (so \( y=-4 \))
2. At \( x=-4 \): \( y=5 \) (so \( y=5 \))
3. At \( x=-3 \): \( y=0 \) (so \( y=0 \))
4. At \( x=-2 \): \( y=-5 \) (so \( y=-5 \))
5. At \( x=5 \): \( y=1 \) (so \( y=1 \))

So the \( y \)-values are \( -5, -4, 0, 1, 5 \), which is the second option: \(\{-5, -4, 0, 1, 5\}\)

Answer:

\(\{-5, -4, 0, 1, 5\}\) (the second option, which is \(\{-5, -4, 0, 1, 5\}\))