How do we determine if a given graph represents a function?

We can determine if a given graph represents a function by using the vertical line test.

The Vertical Line Test Explained

The vertical line test is a visual method used to determine whether a relation, represented as a graph, is a function.

The Rule: If any vertical line drawn on the graph intersects the graph at more than one point, then the graph does not represent a function.
Why it Works: A function, by definition, assigns each input (x-value) to exactly one output (y-value). If a vertical line intersects the graph at two or more points, it means that for that particular x-value, there are multiple corresponding y-values. This violates the definition of a function.
Conversely: If a vertical line drawn anywhere only intersects the graph at only one spot, this means that each x value corresponds to only one y value, so the graph represents a function.

Visualize or Draw Vertical Lines: Imagine drawing vertical lines across the entire graph.
Check for Intersections: Observe the points where the vertical lines intersect the graph.

Determine if it's a Function:

Condition	Conclusion
No vertical line intersects the graph at more than one point.	The graph represents a function.
At least one vertical line intersects at more than one point.	The graph does not represent a function.

Let's consider a few examples:

Example 1: A straight line (non-vertical) A straight line that is not vertical will always pass the vertical line test. Any vertical line will intersect the graph at only one point. Therefore, a non-vertical straight line represents a function.
Example 2: A parabola opening upwards: A parabola opening upwards also represents a function. Every vertical line will only cross the parabola at most once.
Example 3: A circle: A circle does not represent a function. A vertical line drawn through the center of the circle will intersect the circle at two points. This means there's an x-value with two corresponding y-values, thus violating the definition of a function.
Example 4: A vertical line: A vertical line also does not represent a function. In this case, every vertical line (including the line itself!) will intersect the graph at infinitely many points.