Functions are relations that derive one output for each input, or one y-value for any x-value inserted into the equation. For example, the equations y = x + 3 and y = x^{2} - 1 are functions because every x-value produces a different y-value. In graphical terms, a function is a relation where the first numbers in the ordered pair have one and only one value as its second number, the other part of the ordered pair.

## Examining Ordered Pairs

An ordered pair is a point on an x-y coordinate graph with an x and y-value. For example, (2, -2) is an ordered pair with 2 as the x-value and -2 as the y-value. When given a set of ordered pairs, ensure that no x-value has more than one y-value paired to it. When given the set of ordered pairs [(2, -2), (4, -5), (6, -8), (2, 0)], you know that this is not a function because an x-value -- in this case -- 2, has more than one y-value. However, this set of ordered pairs [(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)] is a function because a y-value is allowed to have more than one corresponding x-value.

## Solving for Y

It is relatively easy to determine whether an equation is a function by solving for y. When you are given an equation and a specific value for x, there should only be one corresponding y-value for that x-value. For example, y = x + 1 is a function because y will always be one greater than x. Equations with exponents can also be functions. For example, y = x^{2} - 1 is a function; although x-values of 1 and -1 give the same y-value (0), that is the only possible y-value for each of those x-values. However, y^{2} = x + 5 is not a function; if you assume that x = 4, then y^{2} = 4 + 5 = 9. y^{2} = 9 has two possible answers (3 and -3).

## Vertical Line Test

Determining whether a relation is a function on a graph is relatively easy by using the vertical line test. If a vertical line crosses the relation on the graph only once in all locations, the relation is a function. However, if a vertical line crosses the relation more than once, the relation is not a function. Using the vertical line test, all lines except for vertical lines are functions. Circles, squares and other closed shapes are not functions, but parabolic and exponential curves are functions.

## Using an Input-Output Chart

An input-output chart displays the output, or result, for each input, or original value. Any input-output chart where an input has two or more different outputs is not a function. For example, if you see the number 6 in two different input spaces, and the output is 3 in one case and 9 in another, the relation is not a function. However, if two different inputs have the same output, it is still possible that the relation is a function, especially if squared numbers are involved.