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.

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.

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).

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.

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.