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.

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.

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.