A function is a special mathematical relationship between two sets of data, where no member of the first set is directly related to more than one member of the second set. The easiest example to illustrate this is grades in school. Let the first set of data contain every student in a class. The second set of data contains every possible grade a student could receive. In order to satisfy the mathematical definition of a function, each student must receive exactly one grade. Not all the grades may be given, and some may be given more than once--for example, more than one student might get a 95 percent final grade. But no student receives more than one grade. The best way to find out whether an equation represents a function or not is by graphing the equation and then applying the vertical line test.
- Graph paper
Graph the two-variable equation on graph paper. For a straight line this means graphing two or more points on the line and connecting the dots. Methods for graphing other shapes may vary: Sometimes you can recognize the specific shape, and how to graph it, from its equation. Sometimes you just have to graph many points from the equation, selecting an x-value, finding the corresponding y-value and plotting that point on the graph. Then select a new x-value, find its corresponding y-value, graph that point, and continue on until you can get a feel for the shape.
Draw a vertical line through any given point on the line or lines you graphed. Does it cross through the graph you drew at one point, or at more than one point? If it crosses through the graph at more than one point, this proves that the equation you're considering is not a function.
Imagine running the vertical line you drew all the way to the left and all the way to the right of the graphed equation. Would it, at any point along the graph at all, intersect the lines at more than one point at once? If the answer is no, you've identified a function. If it's yes, you have proven that the equation does not represent a function.