Functions

A function $f:\,A\rightarrow{B}$ is a relation $f:\,A\rightarrow{B}$ which is everywhere defined and not one-many.

• $dom(f)=A=preran(f)$

Inverse

For a function $f:\,A\rightarrow{B}$ to have its inverse relation $f^{-1}:\,B\rightarrow{A}$ be also a function:

• $f$ is onto
• $f$ is not many-one (in other words, $f$ must be one-one)

The above statement is true for all unrestricted function $f$ that has an inverse $f^{-1}$:

$f(f^{-1}(x))=x=f^{-1}(f(x))=x$

Real-Valued Functions

When both domain and codomains of a function are subsets of $\mathbb{R}$, the function is said to be a real-valued function.

Odd and Even Functions

Odd Function

A function $f$ that is $f(-x)=-f(x)\; \forall x \in \text{Dom}(f)$.

Even Function

A function $f$ that is $f(-x)=f(x)\; \forall x \in \text{Dom}(f)$.