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Internet Resources for the Calculus Student

Relations and Functions 

One way of specifying a relationship between two variable quantities is just by listing all of the related (ordered) pairs (eg for the relation x^2+y^2=25, if we restrict to non-negative integer values for x and y, then the only possibilities for (x,y) are (0,5),(3,4),(4,3),and (5,0)).  But if there are infinitely many posibilities (as in the above example if we let x and y be real numbers that may or may not be integers) then this will not suffice. In such cases involving whole ranges of real number values, a good way of illustrating the relationship is by graphing the set of points whose Cartesian coordinates satisfy the relation. (eg the set of all points (x,y) for which the above relation x^2+y^2=25 holds, is a circle; for y=x it is the diagonal straight line; and for x>y it is all points below that line).

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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