Function vs. Relation
The Vertical Line Test is a method used to determine whether a relation is a function or not. When applied to a graph, if any vertical line intersects the graph in more than one point, then the relation is not a function. On the other hand, if every vertical line intersects the graph at most once, then the relation is a function.
To illustrate this, let's consider two examples:
Example 1: Relation that is a Function
Suppose we have a relation where each x-value corresponds to a unique y-value. Here's a graph of such a relation:
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In this case, we can see that every vertical line intersects the graph at most once. Hence, this relation satisfies the Vertical Line Test and is a function.
Example 2: Relation that is Not a Function
Now, let's consider a relation where one or more x-values have multiple corresponding y-values. Here's a graph of such a relation:
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In this case, if we draw a vertical line passing through the graph, it intersects it at two points. Therefore, this relation fails the Vertical Line Test and is not a function.
Regarding the statement "Every relation is a function, but not every function is a relation," it is false. The correct statement is: "Every function is a relation, but not every relation is a function." This is because a function is a specific type of relation where each input value (x-value) is associated with exactly one output value (y-value). In other words, a function is a relation that passes the Vertical Line Test. However, a relation may not be a function if it fails the Vertical Line Test by having multiple y-values for a single x-value.