Method

Determining if a Relation Is a Function Using Mapping Diagrams

Given a relation, a mapping diagram can be used to determine whether the relation is a function. For example, consider the relation given by the following set of coordinates. {(4,3),(0,-5),(-3,3),(8,0),(4,-5)} To figure out if a relation is a function, there are four steps to follow.

List the Inputs

Start by writing all the inputs of the relation in a column. The inputs are the first components of each pair if the relation is given as a set of coordinates. {( 4,3),( 0,-5),( -3,3),( 8,0),( 4,-5)} Notice that the number 4 is an input for two different points. In the mapping diagram, it is written only once because multiple arrows can be drawn from it. Therefore, the inputs for the given relation are 4, 0, -3, and 8.

If the relation is given as a vertical table, the inputs are the values in the first column.

List the Outputs

Write all the outputs in a second column. When the relation is given as a set of coordinates, the outputs are the second components of each pair. {(4, 3),(0, -5),(-3, 3),(8, 0),(4, -5)} The number - 5 is an output for two different pairs. In the mapping diagram, it is written only once because multiple arrows can be drawn to it. For the given relation, the outputs are 3, -5, and 0.

If the relation is given as a vertical table, the outputs are the values in the second column.

Connect Inputs and Outputs

Draw an arrow from each input to its corresponding output.

Look For Arrows With the Same Tail

Check if there are multiple arrows with the same starting input. In other words, identify inputs that have multiple outputs.

Here, the input 4 has two different outputs, 3 and -5. By definition, for a relation to be a function, every input must be assigned to exactly one output. As such, the given relation is not a function.

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