A math course on usual functions in second grade is always beneficial and allows you to progress well throughout the year.

This math course on usual functions involves the following concepts:

• definition of a numerical function;
• image and antecedent;
• set or domain of definition of a function;
• affine and linear function;
• square function;
• square root function;
• inverse function;
• power function.

## I. Linear functions :

### 1.Definition:

Definition:

A linear function is any function defined by :

where a is a given real.

### 2. graphic representation

Ownership:

In an orthonormal frame of reference, the graphical representation of a linear function defined on by is the line D of equation passing through the origin of the frame of reference (a is a given real number).

Properties:
• If a = 0, the linear function is the null function on , we have for all x, f(x)=0.
• If a>0 , the linear function is strictly increasing on .
• If a<0 , the linear function is strictly decreasing on .

### 3.Characteristic property of linear functions

Ownership:

If f is a linear function, then whatever the real numbers m and p, the rate of change between m and p is constant.

More precisely, if , then, whatever the real m and p are: .

This constant number a is the directing coefficient of the line D representative of the function f.

## II. Affine functions:

### 1.Definition:

Definition:

We call affine function, any function defined by :

where m and p are given real numbers.

### 2. graphic representation

Ownership:

In an orthonormal plane, the graphical representation of an affine function defined on by is the line D of equation where m and p are given real numbers.

Ownership:
• If m = 0, the affine function is a constant function on , we have for all x, f(x)=p.
• If m>0 , the affine function is strictly increasing on .
• If m<0 , the affine function is strictly decreasing on .

### 3.Characteristic property of affine functions

Ownership:

If f is an affine function, then whatever the real numbers a and b are, the rate of change between a and b is constant.

More precisely, if , then, whatever the real numbers a and b are: .

This constant number m is the directing coefficient of the line D representative of the function f.

The number p is called theintercept. We have p=f(0).

### 4. particular affine functions:

Properties:

If p=0 then the affine function is linear.

In this case f(x) is proportional x (m is the proportionality coefficient).

The graphs of linear functions are straight lines that pass through the origin of the reference frame. Their equation is: y=mx.

If m=0 then the affine function is constant. We have for all x, f(x)=p.

The graphs of constant functions are lines parallel to the x-axis. Their equation is: y=p.

## III. The square function:

### 1.Definition:

Definition:

We call square function, any function defined by :

.

### 2. graphic representation

Properties:

In an orthonormal plane, the graphical representation of the square function defined on by is the straight parabola of equation .

Ownership:
• The square function is strictly increasing on .
• The square function is strictly decreasing on .

## IV. The cube function:

### 1.Definition:

Definition:

We call a cube function, any function defined by :

.

### 2. graphic representation

Ownership:

In an orthonormal plane, the graphical representation of the cube function defined on by is the curve of equation .

Ownership:
• The cube function is strictly increasing on .
• The cube function is strictly increasing on .

## V. The inverse function:

### 1.Definition:

Definition:

We call inverse function, any function defined by :

.

### 2. graphic representation

Ownership:

In an orthonormal plane, the graphical representation of the inverse function defined on by is the curve of equation .

Ownership:
• The inverse function is strictly decreasing on .
• The cube function is strictly decreasing on .

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