Function (elementary mathematics)

In Mathematical elementary, more than 90% of the functions met is numerical functions, but the concept of function is not limited to this one.

the article which follows presents some rules to be known on the functions
the article numerical function treats numerical functions in elementary mathematics
the article Fonction (mathematics) presents them in their general information.

Functions

The functions are tools. To be a function should be complied with scrupulous rules. They have many properties. We present here base.

Scrupulous rules

To have a starting whole containing the whole of definition of the function and a whole of arrival.

With each element of this unit definition to make correspond one of those of the whole of arrival.

In elementary mathematics, the first of these rules (however paramount) is often forgotten by the pupils because the examples which are proposed limit to some whole starting and natural arrival in the context of work (together of realities for the numerical functions, together of the points of the plan for the specific functions)

However it remains important .

Image, antecedent

So with an element has , one makes correspond an element B ,

the element B is called the image of has
the element has is called an antecedent of B

Example: F (X) = √x-1 With: x=0, 1, 4, 9 => F (0) =-1, F (1) =0, F (4) =1, F (9) =2

one will say

1 is the image of 4 by F

9 is an antecedent of 2 by F

Examples

A function is thus a tool having a starting whole and a whole of arrival, making correspond to the elements of the first of the elements of the second.

Example 1

A function makes it possible to transform a real number into another for example, by applying a succession of operations to him which must remain identical for each number.

In this case the starting whole is $\ mathbb {R}$ , the whole of definition is the whole of realities for which one can apply the continuation of operations, and the whole of arrival is $\ mathbb {R}$ .

If the continuation of operations consists in adding to raise squared, to remove 4 and take the reverse, one creates a function

Ensemble starting: $\ mathbb {R}$ \ {- 2; 2}
Field of definition: all realities different from 2 and -2
Together of arrival: $\ mathbb {R}$
Correspondence: with $x$ , one associates $\ frac {1} {x^2-4}$

the image of 3 is 1/5 the antecedents of 1/5 are 3 and -3

Example 2

A function also makes it possible to associate points with other points starting from geometrical considerations.

In this case the starting whole is the whole of the points of the plan (or space), the whole of arrival is the whole of the points of the plan (or of space)

For example if has and B are two different points, one can associate, at any point M not located on (AB) the point NR such as AMBN is a parallelogram

Ensemble starting $\ mathcal P$
Ensemble of definition $\ mathcal P - (AB)$
Ensemble of arrivée $\ mathcal P$
Correspondance: with M, one associates NR such as AMBN is a parallelogram.

Name

The function often used end up bearing specific names (sin, cos…), the others are called F, G, etc The image of an element has for the function F is then noted F (a) .

Notation

To summarize all this information, one uses the following writing

\ begin {matrix} F: & D & \ rightarrow have \ \ & X & \ mapsto there \ \ \ end {matrix}

with there = F (X) where D represents the starting whole and has the whole of arrival.

The research of the field of definition, if this one is smaller than the starting whole, remains to make.

Sometimes, one is satisfied with the abusive writing there = F (X) by forgetting all the remainder. What can lead sometimes to dangerous confusions if one simplifies too quickly, as announced it Gottlob Frege in What a function  ? .

subtle example: the function $f: X \ mapsto y=x^2/x$ and the function $g: X \ mapsto y=x$ are not the same ones; since F is “prohibited for x=0 ”.

Random links: