Invertible Functions

Identity Function

Let $X$ be a set. The identity function with domain and codomain $X$, is:

$id:X\to X ~~~~~~~ id(x)=x$

(For all $x\in X$)

Inverse Function

Let $f:X\to Y$ be a function.

• $f$ is invertible if $\exist g:Y \to X, f \circ g = id \land g\circ f = id$
• $g$ is an inverse of $f$
• $f^{-1} = g$
• $T:\R^n\to\R^m$ is invertible $\Leftrightarrow \text{nullity}(T) = 0 \land \text{rank}(T)=m$
• $T:\R^n\to\R^m$ is invertible $\Leftrightarrow m =n$
• The inverse of linear transformation $T$ is also linear.

One-to-one (Injective) Function

Let $f:X\to Y$ be a function.

• $f$ is one-to-one if distinct inputs of $f$produces distinct outputs.
• $f$ is one-to-one if $f(x)=f(y)\implies x = y$
• $f$ is one-to-one if $\forall y \in Y,f(x)=y$ has exactly one solution
• $T$ is one-to-one $\Leftrightarrow \text{nullity}(T)=0$
• $T:\R^{m>n}\to\R^n$ cannot be one-to-one