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.
(4 points)

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 one-to-one if no horizontal line intersects the graph more than once.

Looking at the graph you described, we have two line segments:

1. A line segment from \((-1, 6)\) to \((1, 3)\).
2. A line segment from \((2, 3)\) to \((4, 1)\).

Now, let's analyze each segment:

• The first segment descends from left to right, starting at \(( -1, 6 )\) and ending at \(( 1, 3 )\). This portion does not have any horizontal lines intersecting it more than once.

• The second segment starts at \(( 2, 3 )\) and ends at \(( 4, 1 )\). This segment also descends, and any horizontal line drawn above \(y = 3\) (upward from this segment) will not intersect the graph more than once, except at the point \((2, 3)\).

However, both line segments meet at \((2, 3)\). This means that for the \(y\)-value of \(3\), there are two \(x\)-values (\(1\) from the first segment and \(2\) from the second segment) that give the same output. This shows that the function does not pass the Horizontal Line Test, as the horizontal line \(y = 3\) intersects the graph at more than one \(x\)-coordinate.

Thus, we conclude that the function is not one-to-one because there exist two distinct inputs (specifically \(x = 1\) and \(x = 2\)) that produce the same output (\(y = 3\)). Therefore, it fails the Horizontal Line Test.