Relations and Functions
Define a relation as any set of ordered pairs and a function as a relation where each input maps to exactly one output. Use the vertical line test on graphs and analyze tables and mapping diagrams.
A lesson of Algebra 1, in Functions & Relations, about 25 minutes.
By the end of this lesson, you will be able to say precisely what makes a relation a function, decide it from ordered pairs, a table, a mapping or a graph, and use the vertical line test and explain why it works.
One Input, One Answer
A relation is any pairing of inputs with outputs. That is all. A list of ordered pairs is a relation, a table is a relation, a graph is a relation.
A function is a relation with one rule added, and the rule is the whole lesson:
Every input has exactly one output.
Not "one input per output". Not "no repeats anywhere". One output per input.
Here is why that rule is worth having. A vending machine where pressing B4 sometimes gives chips and sometimes gives a candy bar is broken. You could still describe what it does, but you could not rely on it. A function is the machine that is not broken: the same input always produces the same answer.
$$\text{input} \longrightarrow \boxed{\text{rule}} \longrightarrow \text{one output}$$
Everything built on top of functions, which is most of the mathematics you have left, depends on that reliability.
Outputs Are Allowed to Repeat
The rule catches people from the wrong direction, so be precise about it.
| Relation | A function? | Why |
|---|---|---|
| ${(1,4), (2,5), (3,6)}$ | yes | Three inputs, each with one output |
| ${(1,4), (2,4), (3,4)}$ | yes | The OUTPUT repeats, and that is allowed |
| ${(1,4), (1,5), (2,6)}$ | no | The input $1$ has two different outputs |
| ${(1,4), (1,4), (2,5)}$ | yes | $1$ is listed twice with the SAME output, so it still has one |
The second row is the one to remember. Every student in a class having the same birthday month is fine. One student having two different birthday months is not.
The test, in words: find any input that appears twice. If its outputs disagree, it is not a function. If they agree, it is.
Sort It: Function or Not
Each relation below is a set of ordered pairs. Put it in the right group.
Example 1: Reading It Off a Table
Here is the whole method on one table, worked all the way through.
| $x$ | $y$ |
|---|---|
| $-1$ | $3$ |