A function is a system by which the elements of one set are all assigned to exactly one element of another set. A function may take real numbers and, according to some rule, assign them all to an integer value. A function like this might, for example, round every real number up to the nearest integer. Thus, 1.2, 1.009, and 2 would all be rounded up to 2. The set of real numbers is called the domain of this function, and the set of integers is called the range. The elements of the domain are the inputs of the function, and the elements of the range are the outputs. To go from an input to an output, a rule is needed--in this case, the rule is that every real number shall be rounded up to the nearest integer.

Every function has these three parts: a domain, a range, and a rule. A
function is named by a single letter. If the function *f*, for example, assigns
each element in the set *S* a correspondence with a unique element in the set
*T*, then it is written *f* : *S*â√ú’*T*. In this case, *S* is the domain
of *f*, and *T* is the range of *f*. All that is left for *f* is a rule by
which the correspondence between *S* and *T* is made. For the sake of
simplicity, let *S* and *T* be the same set: real numbers (often the domain and
range of a function are the same). Let the rule by which the function *f*
assigns a correspondence between *S* and *T* be that every member of *S* is
doubled to be a member of *T*. Then, the rule can be written this way: *f* (*x*) = 2*x*, where *x* is any element of *S*. Hence, for a given element of *S*, its
corresponding element in *T* has twice the value.

It is important that in a function every input is assigned to exactly one output. That is, every element in the domain of a function must have one and only one corresponding element in the range of that function. The purpose of a function is to assign a value from another set (the range) to each value in a given set (the domain), so if there were more than one element of the range that corresponded to one element in the domain, the function would be ambiguous, and useless. It is acceptable, however, if more than one element in the domain corresponds to the same element of the range. When this happens, every element of the domain still has one and only one counterpart in the range. The following diagram might make these concepts more clear. It is a conceptual illustration of a function.

Figure %: A function *f* assigns each element of its domain, *S*, to a unique
element of its range, *T*.

The trigonometric functions have different domains and different ranges. The rule for trigonometric functions is different for each function, and depends on certain ratios created by the terminal and initial sides of the angle. In the next section the trigonometric functions will be defined.

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