Points Which of the following best describes the graph below? +3 +2 +1 5 -4 -3 -2 -1 1 2 3 4 -1 -2 -3 -4 A. It is not a function. B. It is a many-to-one function. C. It is a function, but it is not one-to-one. D. It is a one-to-one function. ? PREVIOUS
The Correct Answer and Explanation is:
The correct answer is: D. It is a one-to-one function.
Explanation
To analyze the given graph, we first determine if the red curve represents a function. A graph represents a function if it passes the vertical line test—that is, no vertical line should intersect the graph at more than one point. This condition ensures that each input (x-value) corresponds to exactly one output (y-value).
Looking at the graph, each vertical line would intersect the curve at only one point. Therefore, it is a function.
Next, we examine whether it is a one-to-one function. A function is one-to-one if it also passes the horizontal line test—meaning no horizontal line intersects the graph more than once. This guarantees that each output (y-value) is mapped from exactly one input (x-value), which means the function is injective.
From the graph provided, any horizontal line would also intersect the curve at most once. This means that no two different x-values produce the same y-value. Hence, the function passes both the vertical and horizontal line tests.
This confirms the function is one-to-one (injective) and well-defined (it assigns exactly one output to each input).
Let’s briefly address the other answer choices to show why they are incorrect:
- A. It is not a function – Incorrect, because it passes the vertical line test.
- B. It is a many-to-one function – Incorrect, since each output is matched with only one input, not multiple.
- C. It is a function, but it is not one-to-one – Incorrect, because the function is one-to-one as it passes the horizontal line test.
- D. It is a one-to-one function – Correct, for the reasons described above.
Thus, the graph best describes a one-to-one function.
