Does this graph show a function? Explain how you know.
A. Yes; the graph passes the vertical line test.
B. No; there are y-values that have more than one x-value.
C. Yes; there are no y-values that have more than one x-value.
D. No; the graph fails the vertical line test.
The Correct Answer and Explanation is :
The correct answer is A. Yes; the graph passes the vertical line test.
Explanation:
A function is a mathematical relationship where each input (or x-value) is associated with exactly one output (or y-value). To determine whether a graph represents a function, one key criterion is the vertical line test.
The vertical line test is applied by drawing vertical lines through various points of the graph. If at any point, a vertical line crosses the graph at more than one location, then the graph does not represent a function. This would imply that a single x-value corresponds to multiple y-values, which violates the definition of a function.
On the other hand, if no vertical line intersects the graph at more than one point, then for each x-value there is exactly one y-value, meaning the graph does represent a function. This is why the graph passes the vertical line test.
Let’s explore the other options for clarity:
- B. No; there are y-values that have more than one x-value. This is incorrect because it suggests that a y-value has multiple corresponding x-values. While this is common in functions, it doesn’t affect the determination of whether the graph is a function.
- C. Yes; there are no y-values that have more than one x-value. This is not necessarily true. A function can still have y-values associated with different x-values (i.e., many-to-one relationships), but each x-value must map to only one y-value. This is not the defining feature of whether the graph represents a function.
- D. No; the graph fails the vertical line test. This is incorrect if the graph passes the vertical line test, as mentioned earlier.
Thus, when a graph passes the vertical line test, it confirms that for every x-value, there is exactly one corresponding y-value, making the graph a representation of a function.