Skill: Distinguish relations from functions

A relation pairs inputs with outputs, usually written as ordered pairs $(x,y)$. It is a function only when each input has exactly one output.

Worked example

Consider the relation

$$R=\{(0,0),(1,1),(1,-1),(4,2),(4,-2)\}.$$

Step 1: Identify the inputs.
The inputs are the first coordinates:

$$0,\ 1,\ 1,\ 4,\ 4$$

Step 2: Check repeated inputs.
The input $1$ is paired with both $1$ and $-1$:

$$(1,1)\quad\text{and}\quad(1,-1)$$

The input $4$ is also paired with two different outputs:

$$(4,2)\quad\text{and}\quad(4,-2)$$

Step 3: Decide whether it is a function.
Because some inputs have more than one output, $R$ is not a function. It is still a relation.

On a graph, this same conclusion comes from the vertical-line test: a vertical line at $x=1$ or $x=4$ would cross the graph twice. Any graph that a vertical line crosses more than once does not represent a function.

Remember that different inputs may share the same output. For example, $(2,5)$ and $(3,5)$ would not violate the function rule. The rule only prohibits one input from having multiple outputs.