# 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 **only 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/