# Functions

**Relations, Domain and Range**

** **

A relation refers to when a value is input into something, like an equation, and an output is found. The x-y t-chart in the last lesson would be an example of a relation. We input the x-values and used the equation to find the y-values.

** **

In a relation, the values that are input are called the domain and the outputs that are found are called the range.

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**Identifying a Function of x**

* *

A function is a relation where each input in the domain has exactly one output in the range. The t-chart from above for the equation y = 2x - 1, each value in the domain has one output in the range:

x |
y |

0 |
-1 |

1 |
1 |

2 |
3 |

So, we can say that this relation is a function of x, or f(x)

When is a relation not a function? When a value in the domain has more than one possible output in the range. Take, for example, the equation y^{2} = x. If x is 4, then y could be 2. y could also be -2. Therefore, this is not a function of x.

x |
y |

4 |
-2 |

4 |
2 |

** **

**Vertical Line Test**

** **

When relations are represented on a graph, there is a simple method for determining if they are a function. Every value for x must have only one value for y. You can move a vertical line across the graph. If at any point the line touches more than one point, or touches a line in two places, then that value for x has more than one value for y.

In this set of examples some of the graphs are functions and some are not

The blue line is a function; a vertical line moving across the page would only hit it one time, which means that every x-value has only one y-value.

The green line is not a function; a vertical line would hit it twice.

The red line is a function

The purple line is not a function; all y values are valid outputs for the x-value of 2.