# Identifying Functions

To identify if a relation is a function, we need to check that every possible input has one and only one possible output.

If \(x\) coordinates are the input and \(y\) coordinates are the output, we can say \( y\) is a function of \(x.\)

More formally, given two sets \( X \) and \( Y \), a **function** from \( X \) to \( Y \) maps each value in \( X \) to exactly one value in \( Y \).

This wiki specifically addresses the question of if a particular relation is a function; many more details about functions in general are here.

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## Determining if something is a function graphically

If the function is graphically represented where the input is the \(x\)-coordinate and output is the \(y\)-coordinate, we can use the **vertical line test** to determine if it is a function. If any vertical line drawn can cross the graph at a maximum of one point, then the graph is a function. If there is any place a vertical line can cross the graph at two or more points, the graph is not a function.

Functions are processes that take some input and produce some output, where every valid input produces **one** specific output.

If inputs are on the horizontal \(x\)-axis and outputs are on the vertical \(y\)-axis on the graphs below, how many of them are functions?

## Determining if something is a function algebraically

Consider \( x = y^2 .\) It is true that \(x\) is a function of \(y,\) because no matter what value is used for \(y,\) there is only one possible result for \(x.\)

However, is the reverse true: is \(y\) a function of \(x?\) We can try isolating the \(y\) by square rooting both sides to get \( y = \pm \sqrt{x} .\) Notice the plus-or-minus: this means the equation is really two functions. If we have a value of \( x = 4 \) it means \( y\) could be \(2\) *or* \(-2,\) so this is not a function.

**Cite as:**Identifying Functions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/identifying-functions/