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:

\fbox{
\begin{tabular}{cccc}
$f:$ & $A$ & $\longrightarrow$ & $B$ \\
\quad & $x$ & $\mapsto$ & $y$
\end{tabular}}

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:

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

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:

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

With the settings of Example 1.4:

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

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

Noah Dana-Picard 2007-12-28