Mathematics Study Notes: Functions and Inverse Trigonometric Functions

Cartesian Product Definition: For two non-empty sets $A$ and $B$, the cartesian product $A \times B$ consists of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

• Formula: $A \times B = {(a, b) \mid a \in A, b \in B}$.
• Example: If $A = {1, 2, 3}$ and $B = {p, q}$, then $A \times B = {(1, p), (1, q), (2, p), (2, q), (3, p), (3, q)}$.
• Key Properties:
• If either $A$ or $B$ is empty, then $A \times B = \emptyset$.
• If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$.
• Relations: A relation $R$ from set $A$ to $B$ is any subset of $A \times B$.
• If $(a, b) \in R$, then $b$ is the image of $a$, and $a$ is the preimage of $b$.
• The number of relations from $A$ to $B$ is $2^{mn}$ if $n(A) = m$ and $n(B) = n$.

Definition of Functions: A relation $R$ from $A$ to $B$ is a function if every element in $A$ connects to one element in $B$, denoted as $f : A \rightarrow B$.

• Terminology: If $a \in A$ is linked to $b \in B$, then $b$ is the "image of $a$" under $f$, and $a$ is the "pre-image of $b$". This is written as $b = f(a)$.
• Set Condition: A function is part of $A \times B$ where each $a \in A$ appears once as the first part of a pair.
• Methods of Representation:
• Ordered Pairs: Contains all $a \in A$ and ensures uniqueness of the output.
• Formula Based: Example: $f : \mathbb{R} \rightarrow \mathbb{R}, y = f(x) = 4x$.
• Piecewise Defined: Different rules for different intervals. Example: $f(x) = \begin{cases} x + 1 & -1 \le x < 0 \ 3-x & 0 \le x < 2 \end{cases}$.
• Universal Truth: All functions are relations, but not all relations are functions.