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A course on the generalities of functions with the definition of an antecedent, an image and the study of the representative curve of a function in 3rd grade allows you to progress well. The student will have to know how to calculate the image of a number by a function but also to determine an antecedent by calculation or by exploiting the representative curve of the function.
We will end this lesson with concrete examples from everyday life in ninth grade.
I. Generalities on numerical functions
1. concept of function
A function is a mathematical process that associates to any number x of a starting set
a single number, denoted f(x).
The number x is called the antecedent of the number f(x).
The number f(x) is called the image of the number x by the function f.
We note the function 
.
Example:
We call f the function which, to the length of the side of a square, associates the perimeter of the square.
The function f associates to the number 5, the number 20.
More generally, it associates to the number x, the number 4x.
We note or f(x)=4x.
Remark:
For a function f, we use the notation which reads “f is the function which, to x, associates the number f(x)”.
2.image and antecedent
Remark:
The image of a number is unique. On the other hand, a number can have several antecedents.
Examples:
- Let f be the function such that f(-2)=0.
- the image of -2 by f is 0.
- 0 is an antecedent of -2 by f.
2. Let be the function .
This means that to any number, here noted x, the function f associates a unique number noted f(x).
We say that the image of the number x by the function f is the number x²-4.
- .
- .
3. Table of values
Consider a function .
We can summarize the images and the corresponding priors in a value spreadsheet.
Example:
Consider the function f defined by f(x)=x²-4.
Here is a table of values for this function:
On the first line, we have the antecedents and on the second, the images.
.
Moreover, we can notice that 5 has at least two antecedents which are x = -3 and x = 3.
II. Graphical representation of a numerical function
Consider the plane with an orthonormal reference frame and a function .
The set of points is called the representative curve of the function f and noted .
Example:
The curves below represent the function .
The image of 1.5 by the function f is – 1.75.
- The antecedents of -3 are x=1 and x=-1.
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