Generalities.

Definition 2.1.1   Let $A$ and $B$ be two sets. An application (or a mapping) $f$ from $A$ to $B$ is defined by a subset $G \subset A \times B$ such that $\forall x \in A, \; \exists ! y \in B \; \vert (x,y) \in G$ . We denote $y=f(x)$ .

Notation:

Given an application $f:A \longrightarrow B$ : $A=$ the domain of $f$ and $B=$ the range of $f$ . $y=f(x) \quad \Longrightarrow \begin{cases} y = \text{the image of } x \\ x = \text{a pre-image of } y \end{cases}$

Remark 2.1.2   Two applications $f$ and $g$ are equal if, and only if, they have the same domain $A$ , the same range $B$ and $\forall x \in A, \; f(x)=g(x)$ .

Definition 2.1.3   Let $A,B,C$ be three sets and $f: \; A \longrightarrow B$ and $g: \; B \longrightarrow C$ be two applications. The composition of $f$ with $g$ is the application $gof: \; A \longrightarrow C$ such that $\forall x \in A, \; (gof)(x)=g(f(x))$ .

Example 2.1.4   Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ and $g: \mathbb{R} \longrightarrow \mathbb{R}$ are given by $f(x)=3x-5$ and $g(x)=7x+2$ , then:

$$\forall x \in \mathbb{R}, \; (gof)(x)=7(3x-5)+2=21x-33.$$

Example 2.1.5   The function $f: \quad \mathbb{R} \longrightarrow \mathbb{R}$ given by $f(x)=\cos (2x+1)$ is equal to $f_2of_1$ , where:

• $f_1: \quad x \mapsto 2x+1$ ;
• $f_2: \quad x \mapsto \cos x$ .

Remark 2.1.6   Generally $gof \neq fog$ , even when both compositions are well-defined.

With the settings of Example 1.4:

$$\forall x \in \mathbb{R}, \; (fog)(x)=3(7x+2)-5=21x+1.$$

Thus $fog \neq gof$ , according to remark 1.2.