Numerical function

When we express that a quantity depends on another quantity we suppose that there exists a means of obtaining this quantity from an other. And if these quantities are represented by variables, then variable is function of another, when there is a rule which makes it possible to obtain the value of this variable, starting from the value of the other.

Example: the " quantity; quantify affaire" on a company depends on the " quantity; Nb of products vendus"

A numerical function is a rule which makes it possible to associate with a reality another real number.
Donnons the example of a grocer who increases the prices of all her articles by 20%. To add at each price 20% of the price, amounts multiplying each price per 120%. The rule which the grocer will apply at each price is the multiplication by 1,2 and we will say that the new price is function of old.

Definition

A numerical function $f$ or real function of a real variable of a $D$ part of $\ mathbb R$ in $\ mathbb R$ , is a correspondence (or application) which with any element $x$ of $D$ associates a reality and only one noted $f (X)$ .
This reality $f (X)$ is the image of $x$ by $f$ .

This $D$ part of $\ mathbb R$ is called the unit of definition of $f$ .

Notation

We note the function:

\begin{matrix} F: & D \ subset \ mathbb R & \ rightarrow & \ mathbb R \ \ & X & \ mapsto & F (X) \end{matrix} (to observe that the second arrow has a push rod that the first does not have)

or more simply

f:  X \ mapsto F (X)

Example

That is to say the function which with any real number of the interval $-1; +1$ associates its square decreased by 1.

We can define the function $f$ following manners:

That is to say $f$ defined by:

for any reality

x

-1; +1, \ F (X) = x^2 - 1

or:

\ begin {matrix} F: & & \ rightarrow & \ mathbb R \ \ & X & \ mapsto & F (X) = x^2-1 \ end {matrix}

Notice

We should not confuse F and F ( X ). In the preceding example F is the rule which raises a reality squared and 1 cuts off to him, while F ( X ) is equal to the reality X ² - 1 which is associated with X .

Together of definition

That is to say F a function of D in

\ mathbb R

.
Soit X a reality. If X belongs to D , then it is said that F is defined in X , and if X does not belong to D it is said that F is not defined in X .

Remarks

the whole of definition of a function can be given in the statement defining the function and if not it must be given.
To seek the whole of definition or the field of definition of a function, it is to determine realities X such as F ( X ) exists.

Traditional errors

Good number of high-school pupils hold for true the relation F (a+b)=f (a)+f (b):

for the function square , that would give (a+b) ² =a ² +b ², which is false (see remarkable identity);
for the function sine , that would thus give sin (a+b)=sin (a)+sin (b) 0=sin (180°) =sin (90°+90°) =sin (90°) +sin (90°) =2, that is to say 0=2 (see goniometrical Function)
for the function logarithm , that would thus give ln (a+b)=ln (a)+ln (b) ln (2) =ln (1) +ln (1) =0, which is still false.

In fact confusion comes from the abusive application of the rules of calculation only valid for the linear functions, in other words for the situations of proportionality.

See too

Function (elementary mathematics)
Function (mathematics)

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