For each graph below, state whether it represents a function.
The Correct Answer and Explanation is :
To determine whether a graph represents a function, we use the vertical line test. This test states that a graph represents a function if and only if every vertical line intersects the graph at most once.
Answer:
- Graph A: Represents a function.
- The graph of a straight diagonal line passes the vertical line test because no vertical line intersects it more than once. Each (x)-value corresponds to exactly one (y)-value.
- Graph B: Does not represent a function.
- This graph of a circle fails the vertical line test because vertical lines passing through the center of the circle (or any point within the circle’s range) intersect the graph at two points. Thus, some (x)-values have multiple corresponding (y)-values.
- Graph C: Represents a function.
- The graph of a parabola (opening upward or downward) passes the vertical line test. Each vertical line intersects the graph at only one point. Each (x)-value maps to a single (y)-value.
- Graph D: Does not represent a function.
- The graph of two intersecting diagonal lines fails the vertical line test. Any vertical line passing through the intersection of the two lines intersects the graph at two points. This implies that some (x)-values have multiple (y)-values.
Explanation:
A function is a relation in which each input ((x)) is associated with exactly one output ((y)). The vertical line test checks this condition visually. For graphs:
- A graph that passes the test (intersects at most once with any vertical line) represents a function because no (x)-value is associated with more than one (y)-value.
- A graph that fails the test does not represent a function since some (x)-values correspond to multiple (y)-values.
In summary, Graphs A and C represent functions, while Graphs B and D do not. Functions are essential for modeling relationships where every input has a unique output, ensuring the graph satisfies the vertical line test.