# 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.

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 y2 = 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.