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:

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

is a function because *y* will always be one greater than *x*. Equations with exponents can also be functions. For example:

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:

is not a function; if you assume that *x* = 4, then

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.