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The terms function, mapping, map and transformation are usually used synonymously. The term operator is frequently used for dyadic functions expressed using infix notation.
Essentially, a function is a "rule" that assigns a unique output to each given input. Here are some examples of functions:
The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is deterministic, always producing the same output from the same input. In this way, a function may be thought of as a " machineA machine is any mechanical or electrical device that transmits or modifies energy to perform or assist in the performance of tasks. It normally requires an input as a trigger, and transmits the modified energy to an output, which performs the desired tas" or a " black boxA black box is a system whose input and output characteristics are well understood, but one has no idea what is going on inside''. There are several main senses: In electronics, a sealed piece of replaceable equipment—see line-replaceable unit . In comput", converting a valid input into a unique output. The input is often called the argument of the function, and the output the value of the function.
A very common type of function occurs when the argument and the function value are both numberA number is an abstract entity used to describe quantity. There are different types of numbers. The most familiar numbers are the whole numbers {0, 1, 2,. denoted by W and the natural numbers {1, 2, 3,. used for counting and denoted by N . If the negatives, the functional relationship is expressed by a formula, and the value of the function is obtained by direct substitution of the argument into the formula. Consider for example
which assigns to any number x its square.
A straightforward generalization is to allow functions depending on several arguments. For instance,
is a function which takes two numbers x and y and assigns to them their product, xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pairAn ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element''. An ordered pair with first element a and second element b is usually written as a b . The notation a b is also us (x, y), then we can interpret g as a function -- the argument is the ordered pair (x, y), and the function value is xy.
In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output the temperature at that location at that time.
We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics.
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