A function is a relation in which each input has only one output. In the relation , y is a function of x , because for each input x (1, 2, 3, or 0), there is only one output y . x is not a function of y , because the inputy = 3 has multiple outputs: x = 1 andx = 2 .

Examples:
\: y is a function of x , x is a function of y .

: y is not a function of x ( x = 3 has multiple outputs), x is a function of y .

: y is a function of x , x is not a function of y ( y = 9 has multiple outputs).

: y is not a function of x ( x = 1 has multiple outputs), x is not a function of y ( y = 2 has multiple outputs).

To determine whether y is a function of x , given a graph of a relation, use the following criterion: if every vertical lineyou can draw goes through only 1 point, y is a function of x . If you can draw a vertical line that goes through 2 points, yis not a function ofx . This is called the vertical line test.

Example 1: In the following graph, y is a function of x :

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A function is one kind of interrelationship among objects. For every finite sequence of objects (called the arguments), a function associates a unique object (called the value). More formally, a function is defined as a set of finite lists of objects, one for each combination of possible arguments. In each list, the initial elements are the arguments, and the final element is the value. For example, the function contains the list , indicating that integer successor of is . A relation is another kind of interrelationship among objects in the universe of discourse. More formally, a relation is an arbitrary set of finite lists of objects (of possibly varying lengths). Each list is a selection of objects that jointly satisfy the relation. For example, the < relation on numbers contains the list , indicating that is less than . Note that both functions and relations are defined as sets of lists. In fact, every function is a relation. However, not every relation is...

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