Question

Explain how the vertical line test is used to detect functions.

Short answer

Answer: The vertical line test is a visual method used to determine whether a given relation is a function or not by checking if there is more than one point of intersection between the graph and any vertical line drawn on the graph. To apply the vertical line test, follow these steps: 1. Plot the graph of the given relation in the Cartesian coordinate system. 2. Draw several vertical lines across the graph, covering the entire graph by placing the lines at regular intervals or visualize the intersections between the graph and the lines. 3. Observe how many times each vertical line intersects the graph. 4. If none of the vertical lines intersect the graph more than once, the graph represents a function. However, if there is even one vertical line that intersects the graph at multiple points, the graph does not represent a function.

Step-by-step solution

Definition of a function

A function is a relation between two sets in which each element of the first set (the domain) is related to exactly one element in the second set (the range).

Purpose of the vertical line test

The vertical line test is a visual method used to determine whether a given relation is a function or not. It can be applied to graphs in the Cartesian coordinate system and checks if there is more than one point of intersection between the graph and any vertical line drawn on the graph. If any vertical line intersects the graph at more than one point, the graph does not represent a function, as this means that there is more than one value associated with the same input.

Plot the graph

In order to apply the vertical line test, you first need to have a graph of the given relation in the Cartesian coordinate system.

Draw vertical lines

Draw several vertical lines across the graph. To be thorough, make sure you cover the entire graph by placing the lines at regular intervals, taking into account the scale of the x-axis. Alternatively, you can also imagine the vertical lines and visualize the intersections between the graph and the lines.

Check for intersections

Observe how many times each vertical line intersects the graph. If any vertical line intersects the graph more than once, this means there is an input in the domain with multiple corresponding values in the range, which violates the definition of a function.

Determine whether it's a function

If none of the vertical lines intersect the graph more than once, the graph represents a function. However, if there is even one vertical line that intersects the graph at multiple points, the graph does not represent a function.