Question

algebra 1 > m.4 identify functions vll look at this graph: graph of a relation on a coordinate plane is this relation a function? yes no

Explanation:

Step1: Recall function definition

A relation is a function if each \( x \)-value has exactly one \( y \)-value. We use the vertical line test: if any vertical line intersects the graph at more than one point, it's not a function.

Step2: Analyze the graph's points

Identify the \( x \)-coordinates of each point. Let's list the points (assuming grid coordinates):

• Leftmost point: \( x = -4 \) (one \( y \)-value)
• Middle left top: \( x = -3 \) (one \( y \)-value)
• Middle left bottom: \( x = -3 \) (another \( y \)-value)
• \( y \)-axis point: \( x = 0 \) (one \( y \)-value)
• Right point: \( x = 3 \) (one \( y \)-value)

Wait, \( x = -3 \) has two different \( y \)-values (from the two points at \( x = -3 \)). Wait, no—wait, let's check again. Wait, the points: let's see the graph. Wait, maybe I misread. Wait, the vertical line at \( x = -3 \) would pass through two points? Wait, no, let's re-express. Wait, the points: let's list their \( (x,y) \):

• Let's assume the grid: the leftmost dot: \( x = -4 \), some \( y \) (say \( y = 2 \))
• Then a dot at \( x = -3 \), \( y = 1 \)
• Then a dot at \( x = -3 \), \( y = -1 \)
• Then a dot at \( x = 0 \), \( y = 2 \)
• Then a dot at \( x = 3 \), \( y = -1 \)

Wait, so \( x = -3 \) is paired with two different \( y \)-values (\( y = 1 \) and \( y = -1 \))? Wait, no—wait, maybe the \( x \)-coordinates: wait, the vertical line test: draw a vertical line at \( x = -3 \). Does it intersect more than one point? Yes, because there are two points with \( x = -3 \). Wait, but wait—wait, maybe I made a mistake. Wait, no, the vertical line test: if any \( x \) has multiple \( y \), it's not a function. Wait, but wait, let's check again. Wait, the points: let's look at the \( x \)-values. The \( x \)-values are: -4, -3, -3, 0, 3. So \( x = -3 \) is repeated with different \( y \)-values. Therefore, by the definition of a function (each \( x \) maps to exactly one \( y \)), this relation is not a function? Wait, no—wait, maybe I misread the points. Wait, no, the two points at \( x = -3 \): one above the x-axis, one below. So same \( x \), different \( y \). So vertical line at \( x = -3 \) hits two points. Therefore, it fails the vertical line test. Wait, but wait, the original problem's graph: let me re-express. Wait, maybe the \( x = -3 \) points: no, maybe I messed up. Wait, no—wait, the key is: for a function, each input \( x \) has exactly one output \( y \). So if any \( x \) has more than one \( y \), it's not a function. So in this graph, the \( x \)-value of -3 is associated with two different \( y \)-values (the two dots at \( x = -3 \)), so the vertical line test fails. Wait, but wait, maybe I made a mistake. Wait, no—wait, let's check again. Wait, the points: let's list them:

1. \( (-4, 2) \) (leftmost, top)
2. \( (-3, 1) \) (middle left, top)
3. \( (-3, -1) \) (middle left, bottom)
4. \( (0, 2) \) (on y-axis)
5. \( (3, -1) \) (right)

So \( x = -3 \) is paired with \( y = 1 \) and \( y = -1 \). Therefore, this relation is not a function. Wait, but the options are yes or no. So the answer is no? Wait, but wait—wait, maybe I misread the \( x \)-coordinates. Wait, maybe the two points at \( x = -3 \) are not the same \( x \)? No, the vertical line at \( x = -3 \) would pass through both. So the vertical line test shows it's not a function. Wait, but let's confirm the vertical line test: a vertical line intersects the graph at more than one point, so it's not a function.

Answer:

no