Problem 100 - College Algebra

Question

Explain how the vertical line test is used to determine whether a graph is a function.

Step-by-Step Solution

Answer

The vertical line test is used to determine if a graph is a function by checking if any vertical line drawn through the graph intersects at more than one point. If such a line can be drawn, the graph does not represent a function. If no such line exists, the graph does represent a function.

Step 1: Definition of a Function

To apply the vertical line test, it is crucial to understand what a function is. A function is a special type of relation where every input (or element in the domain) is associated with exactly one output (or element in the range).

Step 2: Explanation of the Vertical Line Test

The vertical line test is a visual way to determine if a graph of a relation represents a function by checking if any vertical line drawn through the graph touches the graph at more than one point.

Step 3: Application of the Vertical Line Test

If you can draw any vertical line that intersects more than one point on the relation, then the relation is not a function. If, however, all vertical lines intersect the graph no more than once, the graph does represent a function.

Step 4: Interpreting the Vertical Line Test

This is because a function can only have one output, y, for each unique input, x. If a vertical line intersects the graph more than once, a single x value has multiple y values, which contradicts the definition of a function.