11 of 1111 of 11 Items

Question

Feature
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.

Answer 1

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

In the graph you've described, we see two line segments:

1. The first segment connects the points (-1, 6) and (1, 3).
2. The second segment connects the points (2, 3) and (4, 1).

Now let's analyze these segments with respect to the Horizontal Line Test:

• The first line segment goes from (−1, 6) to (1, 3), and it is decreasing. In this segment, any horizontal line drawn between the y-values of 6 and 3 (e.g., at y=5, y=4, or y=3) will intersect the segment at most once.

• The second segment goes from (2, 3) to (4, 1). This segment is also decreasing. A horizontal line at y=3 will intersect the function at (2, 3) and (1, 3), meaning it intersects the graph at two points.

Since we found a horizontal line that intersects the graph at more than one point (specifically, the horizontal line at y=3 intersects the graph at both (1, 3) and (2, 3)), the function fails the Horizontal Line Test.

Therefore, the function is not one-to-one.