Relations and Functions
A function is a special type of relation - one in which one of the variables is completely determined by the other. Of the above examples with a continuous range of variables, only y=x is a function, for the others there are more than one possible y for each x. The first (restricted) example is a function, but it wouldn't have been if we had allowed negative as well as positive values for y.
A relation between two variables may determine one in terms of the other but not vice versa. E.g. the equation x = y^2 determines x as a function of y, but not y as a function of x, since for x>0 there are two possible values for y (ie + or - the square root of x).
The graph of a relationship between two real variables is the set of all points in a plane whose Cartesian coordinates x and y satisfy the relationship. A set of points that is the graph of a function must satisfy the "Vertical Line Test" - ie each vertical line can meet the graph in at most one point.
If you have come across any other good web-based illustrations of these
and related concepts, please do let
us know and we will add them here.
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