To identify if a relation is a function, we need to check that every possible input has one and only one possible output.
If coordinates are the input and coordinates are the output, we can say is a function of
More formally, given two sets and , a function from to maps each value in to exactly one value in .
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 -coordinate and output is the -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 -axis and outputs are on the vertical -axis on the graphs below, how many of them are functions?
Consider It is true that is a function of because no matter what value is used for there is only one possible result for
However, is the reverse true: is a function of We can try isolating the by square rooting both sides to get Notice the plus-or-minus: this means the equation is really two functions. If we have a value of it means could be or so this is not a function.