Question

which graph shows a set of ordered pairs that represent a function?

Explanation:

To determine which graph represents a function, we use the vertical line test: a set of ordered pairs is a function if no two points have the same \( x \)-value (i.e., no vertical line intersects the graph more than once).

Step 1: Analyze the first graph

Check the \( x \)-values of each point. Let's list the ordered pairs (estimating coordinates):
- \( (-2, 3) \), \( (-1, 2) \), \( (1, 1) \), \( (2, -1) \), \( (3, -3) \), \( (5, -5) \). Wait, no, actually, looking at the first graph, let's check \( x \)-values. Wait, maybe I misread. Wait, the first graph: let's check if any \( x \)-value is repeated. Wait, no, maybe I made a mistake. Wait, let's check the second graph.

Step 2: Analyze the second graph

List the ordered pairs (estimating coordinates):
- \( (-4, -4) \), \( (-2, 2) \), \( (1, 1) \), \( (2, -5) \), \( (4, 4) \), \( (4, -3) \). Wait, here \( x = 4 \) has two \( y \)-values (\( 4 \) and \( -3 \)), so it fails the vertical line test.

Step 3: Analyze the third graph

List the ordered pairs:
- \( (-3, 2) \), \( (-2, 2) \), \( (0, 1) \), \( (2, 3) \), \( (4, -4) \), \( (2, -4) \)? Wait, no, wait \( x = -3 \) and \( x = -2 \) both have \( y = 2 \)? Wait, no, \( x = -3 \) and \( x = -2 \) are different \( x \)-values. Wait, no, the third graph: points at \( (-3, 2) \), \( (-2, 2) \), \( (0, 1) \), \( (2, 3) \), \( (4, -4) \), \( (2, -4) \)? Wait, no, \( x = 2 \) would have two \( y \)-values? Wait, no, maybe I'm misreading. Wait, the fourth graph? Wait, no, the problem has four graphs? Wait, the user's image shows four graphs? Wait, the original question: "Which graph shows a set of ordered pairs that represent a function?" Let's re-express:

Wait, the key is the vertical line test: each \( x \) has at most one \( y \).

Looking at the second graph (middle top? No, the second graph in the first row? Wait, let's check the fourth graph (bottom right). Wait, no, let's check the first graph:

Wait, first graph (left):

Points: Let's list \( x \)-coordinates:

- \( x = -2 \): one point
- \( x = -1 \): one point
- \( x = 1 \): one point
- \( x = 2 \): one point
- \( x = 3 \): one point
- \( x = 5 \): one point

Wait, no, maybe I'm wrong. Wait, the second graph (middle):

Points:

- \( x = -4 \): one point
- \( x = -2 \): one point
- \( x = 1 \): one point
- \( x = 2 \): one point? No, \( x = 4 \): two points ( \( (4, 4) \) and \( (4, -3) \) ), so that's a repeat of \( x = 4 \), so not a function.

Third graph (right top):

Points:

- \( x = -3 \): one point
- \( x = -2 \): one point
- \( x = 0 \): one point
- \( x = 2 \): one point
- \( x = 4 \): one point
- \( x = 2 \): another point? Wait, no, \( x = 2 \) has two points? No, maybe I'm misreading. Wait, the fourth graph (bottom right):

Points:

- \( x = -4 \): one point
- \( x = -2 \): one point
- \( x = 0 \): one point (at \( y = -2 \))

Wait, no, let's check the first graph again. Wait, maybe the correct graph is the second one? No, wait, the second graph has \( x = 4 \) with two \( y \)-values. The third graph: \( x = -3 \) and \( x = -2 \) both have \( y = 2 \), but those are different \( x \)-values, so that's okay. Wait, no, \( x = -3 \) and \( x = -2 \) are different, so same \( y \) is okay. Wait, the key is same \( x \) with different \( y \) is bad.

Wait, let's check the fourth graph (bottom right):

Points:

- \( (-4, 2) \), \( (-2, 1) \), \( (-2, -2) \), \( (0, -2) \). Wait, \( x = -2 \) has two \( y \)-values ( \( 1 \) and \( -2 \) ), so that's bad.

Wait, the first graph (left):

Points:

- \( (-2, 3) \), \( (-1, 2) \), \( (1, 1) \), \( (2, -1) \), \( (3, -3) \), \( (5, -5) \). Wait, no, \( x \)-values are all unique: \( -2, -1, 1, 2, 3, 5 \). So each \( x \) has one \( y \), so it passes the vertical line test.

Wait, maybe I made a mistake with the second graph. Let's recheck the second graph (middle):

Points:

- \( (-4, -4) \), \( (-2, 2) \), \( (1, 1) \), \( (2, -5) \), \( (4, 4) \), \( (4, -3) \). Here, \( x = 4 \) has two \( y \)-values, so it's not a function.

Third graph (right top):

Points:

- \( (-3, 2) \), \( (-2, 2) \), \( (0, 1) \), \( (2, 3) \), \( (4, -4) \), \( (2, -4) \). Wait, \( x = 2 \) has two \( y \)-values ( \( 3 \) and \( -4 \) ), so it's not a function.

Fourth graph (bottom right):

Points:

- \( (-4, 2) \), \( (-2, 1) \), \( (-2, -2) \), \( (0, -2) \). \( x = -2 \) has two \( y \)-values, so not a function.

Wait, so the first graph (left) has all unique \( x \)-values, so it's a function.

Wait, but maybe I misread the graphs. Let's confirm:

The first graph (left) has points with \( x \)-coordinates: \( -2, -1, 1, 2, 3, 5 \) (all distinct), so each \( x \) maps to one \( y \), so it's a function.

The second graph (middle) has \( x = 4 \) with two \( y \)-values, so not a function.

The third graph (right top) has \( x = -3 \) and \( x = -2 \) both with \( y = 2 \), but those are different \( x \)-values, so that's okay? Wait, no, \( x = -3 \) and \( x = -2 \) are different, so same \( y \) is allowed. Wait, no, the problem is same \( x \) with different \( y \). So \( x = -3 \) and \( x = -2 \) are different, so that's fine. Wait, but in the third graph, is there a repeated \( x \)? Let's check the \( x \)-coordinates:

- \( (-3, 2) \): \( x = -3 \)
- \( (-2, 2) \): \( x = -2 \)
- \( (0, 1) \): \( x = 0 \)
- \( (2, 3) \): \( x = 2 \)
- \( (4, -4) \): \( x = 4 \)
- \( (2, -4) \): \( x = 2 \)

Ah! Here \( x = 2 \) has two \( y \)-values ( \( 3 \) and \( -4 \) ), so it's not a function.

The fourth graph (bottom right):

- \( (-4, 2) \): \( x = -4 \)
- \( (-2, 1) \): \( x = -2 \)
- \( (-2, -2) \): \( x = -2 \)
- \( (0, -2) \): \( x = 0 \)

Here \( x = -2 \) has two \( y \)-values, so not a function.

The first graph (left):

- \( (-2, 3) \): \( x = -2 \)
- \( (-1, 2) \): \( x = -1 \)
- \( (1, 1) \): \( x = 1 \)
- \( (2, -1) \): \( x = 2 \)
- \( (3, -3) \): \( x = 3 \)
- \( (5, -5) \): \( x = 5 \)

All \( x \)-values are unique, so it passes the vertical line test and is a function.

Wait, but maybe the second graph (middle) was misread. Wait, no, the second graph has \( x = 4 \) with two points. So the first graph (left) is the function.

Answer:

The first graph (left - most) shows a set of ordered pairs that represent a function.