Chapter 8.1, Problem 25E

To determine

To give:

An example of a relation that is a function, but whose inverse is not a function.

Answer to Problem 25E

Our required relation would be $(1,3),(7,6),(5,9),(4,3),(3,6)$

Explanation of Solution

Given:

To form the inverse of a relation represented by a set of ordered pairs, you switch the coordinates of each ordered pair. For example, the inverse of relation $(1,2),(3,4),(5,6)\text{is (2,1),(4,3),(6,5)}$ .

Calculation:

We know that a relationship will be also a function, when each input in the domain has exactly one specific corresponding output.

We also know that for a relation to be a function, one x- value cannot have two y -values, while two x- values can have same y- value.

To get inverse of function, the function needs to be one-to-one function. In one to one function each out-put comes from only one input that is two x- values cannot have same y- value.

The relation $(1,3),(7,6),(5,9),(4,3),(3,6)$ satisfies our given conditions. Each input corresponds to exactly one out-put.

The inverse of our relation would be $(3,1),(6,7),(9,5),(3,4),(6,3)$ . We can see that 3 corresponds to values, therefore, the inverse of our relation is not a function.

Therefore, our required relation would be $(1,3),(7,6),(5,9),(4,3),(3,6)$ .