Question

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
Use the graph of the function to answer the question.
Is this function one-to-one? Explain how you determined your answer.
(4 points)

Answer 1

To determine if a function is one-to-one, we need to check if each output value (y-value) is associated with a unique input value (x-value). In other words, for a function to be one-to-one, no horizontal line should intersect the graph of the function at more than one point (this is known as the Horizontal Line Test).

Let's analyze the provided segments:

1. The first line segment connects the points \((-1, 6)\) and \( (1, 3)\). This segment moves from a higher y-value to a lower y-value as x increases, indicating a decreasing function over that interval.

2. The second line segment connects the points \( (2, 3) \) and \( (4, 1) \). This segment also moves from a higher y-value to a lower y-value, indicating it is also a decreasing function over that interval.

Now to determine if the function is one-to-one, we can check for overlapping y-values:

- The first segment takes y-values from 6 down to 3.
- The second segment takes y-values from 3 down to 1.

The key observation is that both segments share the y-value of 3 at the point \( (1, 3) \) from the first segment and \( (2, 3) \) from the second segment.

Since there are two different x-values (\(1\) and \(2\)) that correspond to the same y-value (3), the function fails the Horizontal Line Test.

**Conclusion**: The function is not one-to-one because the same output value (3) corresponds to two different input values (\(1\) and \(2\)).