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Let A and B be two nonempty sets. A function from A to B is a rule that assigns to each element in set A, one and only one element of set B. In general, the sets A & B need not be the sets of real numbers. It could be any abstract situation. However, we consider only those functions for which A & B are both subsets of real numbers.
The set A in the above definition is called the domain of the function. We usually denote it by dom f. If x is an element in the domain of a function, then the element that f associates with x is denoted by the symbol f (x) and is called the image of x under f, or the value of f at x. The set of all possible values of f(x), if x varies over the domain is called range of f: If f A→B then the range of f is a subset of B and the set B is called the
co-domain of f.
If x is an element in the domain of a function f the definition of a function requires that f assign one and only one value to x. This means that a function can not be multiple valued. For example the expression ± √x does not define a function of x. Since it assigns two values to each positive x.