What is the definition of a function

Definition of the concept of function

The previous tasks were about dependencies

  • express by a formula and
  • in tabular form

What do the cell phone example and the box example have in common? In both cases we looked at regulations that allow

another, from that
e.g. the length of the side of the square x         the box volume V(x)

assign. This idea of ​​mapping is very important in mathematics. We can think of very different objects (such as natural numbers or real numbers).

We now want to formulate a little more precisely how the idea of ​​assignment can be used for mathematics. To do this, we have to determine which objects are involved for each specific assignment, as is the requirement for precision in mathematics. We summarize all possible "given objects" in a set A. together and imagine that all possible "dependent objects" are in a set B. lie. This is how we get to Definition of the concept of functionas it has been used in mathematics for over 100 years:

definition: Be A. and B. two sets. A function of A. to B. is a prescription that applies to each element of the set A. a (i.e. exactly a) element of the set B. assigns.


  • The amount A. we name Definition set, the amount B. called Target amount.
  • Like other math objects, functions are identified with symbols (usually letters). Let us denote a function of A. to B. with the letter f, so we write for it too

    f : A. B.

    (pronounced: "f is a function of A. to B.").
  • We also say: each xA. will depend on the function f on an element of B.pictured. This element of B. we write as


    Functions are sometimes too Illustrations called.
  • Can be specified by a term (i.e. a formula) how f(x) out x is determined, we speak of a "Term representation the function f". This is for example due to the term representation

    f(x) = x2

    defines the function that assigns its square to each element of the definition set. The statement f(x) = x2 is also known as Function equation designated. Another notation for this is

    f : x x2.

    A term representation is one expressed by a term Assignment rule.

Most electronic arithmetic tools allow a function, i.e. an assignmentregulation to be able to use them later if necessary.