Problem 127 - Precalculus Essentials

Question

Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.

Step-by-Step Solution

Answer

An example of a function that satisfies both conditions of the problem is \(f=\{(1,2), (3,2)\}\). When the ordered pairs are reversed, we have \(f=\{(2,1), (2,3)\}\), which is not a function.

Step 1: Construct a Function Relation

An example of a function containing two ordered pairs where each first element corresponds to one and only one second element is the following: \[ f=\{(1,2), (3,2)\} \] Here, every element in the domain (1, 3) is associated with a single specific element in the codomain (2).

Step 2: Reverse the Components

Now, if reversing the ordered pairs results in \[ f=\{(2,1), (2,3)\} \]

Step 3: Discuss the Reversed Relation

Here, the element 2 in the domain is associated with two different elements in the codomain (1, 3). Therefore, this reversed relation is not a function since a function must follow the rule that any single specific input gives exactly one output.