A graph is shown on the coordinate plane.The x-axis goes from negative 2 to 6 and the y-axis goes from negative 1 to 7. A line segment connects closed points left-parenthesis negative 1 comma 6 right-parenthesis and left-parenthesis 1 comma 3 right-parenthesis. Another line segment connects closed points left-parenthesis 2 comma 3 right-parenthesis and left-parenthesis 4 comma 1 right-parenthesis.

Question

Question
Use the graph of the function to answer the question.
Is this function one-to-one? Explain how you determined your answer.

Answer 1

To determine if the function represented by the graph is one-to-one, we can use the Horizontal Line Test. A function is considered one-to-one if any horizontal line intersects the graph at most once.

From the description of the graph:

1. The first segment connects the points \((-1, 6)\) and \((1, 3)\) and has a negative slope. If we were to draw a horizontal line at any \(y\) value between 3 and 6, it would intersect this line segment only once.

2. The second segment connects the points \((2, 3)\) and \((4, 1)\) and also has a negative slope. If we draw a horizontal line at \(y = 3\), it will intersect both the first segment (at the point \((1, 3)\)) and the second segment (at the point \((2, 3))\). This means that there is a horizontal line that intersects the graph at two different \(x\) values.

Since we found a horizontal line (in this case at \(y = 3\)) that intersects the graph at more than one point, we conclude that the function is not one-to-one.

Summary: The function is not one-to-one because a horizontal line at \(y = 3\) intersects the graph at two different points: \((1, 3)\) and \((2, 3)\).