What is the definition of a function?

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ishpiro eNotes educator| Certified Educator

As described above, a function is a rule that creates correspondence between an element of set x, called domain, and an element of set y, called range. The function is usually denoted as y = f(x), where f(x) means: if one takes x and applies the rule f, y is obtained.

Functions can be given in various forms:

1) Verbal description. For example, one's weekly salary depends on the number of hours worked per week, if one makes $15 an hour.

2) Formula. For example, y = 2x + 3.

3) Table, or set of ordered pairs. For example, {(1, 2), (2, 3), (-1, 4)}. The first number in a pair is an element of x and the second number is an element of y.

4) Graph.

However, not every rule, or every correspondence between an element of set x and an element of set y is a function. In a function, each element of x can correspond to only one element of y. If this is not the case, the correspondence is called a relation, but it is not a function.

For example, the set of ordered pairs{(1, 2), (1, 3), (2, 5)} is NOT a function because for x = 1 there are two values of y: y = 2 and y = 3.

Likewise, the equation `y^2 = x^2` does NOT define a function y = f(x) because it is possible for x = 1 to have two corresponding elements of y: y = 1 and y = -1. On the graph, the curve representing this equation can be crossed by a vertical line in two places.

However, a function CAN have two different elements of x corresponding to one element of y. For example,

{(1, 2), (2, 2), (-1, 3)} is a function and `y = x^2` is a function.


Borys Shumyatskiy eNotes educator| Certified Educator


I suppose you mean the notion of mathematical function. This word has its meaning in different subjects too.

In mathematics, a functions is a rule which takes each element of some set `X` called domain and returns an element from a set `Y` called codomain.

For example, consider number function `f` from [-1, 1] to [-1, 2], `f(x)=x^2.`

Usually a function has one value for each argument, but sometimes many-valued functions are also studied.

For a subset A of a domain, its image is the set of function values on A, i.e. `f(A)={f(x) in Y: x in A}.`

The image of a domain is called a range of a function. In above example it is [0, 1].

If a function f acts from a set X into a set Y, and a function g from Y to Z, then their composition gof is defined. It acts from X to Z as `g(f(x)).`

Much more may be said about functions.