Composition of Functions

Definition

Remember that functions map sets $g:A\to B$

$f:B\to C$

The composition of functions $f\circ g:A\to C$

$f\circ g(a)=f(g(a))$

Domain of $f\circ g$ = Domain of $g$

Codomain of $f\circ g$ = Codomain of $f$

Codomain of $g\subseteq$ domain of $f$

Range of $f\circ g=$ The image of range of $g$ under the function $f$

Range of $g=S$ (subset of codomain of $g$)

Matrix Multiplication can also be understood as a composition of functions.

Function

$$\text{Codomain of F is}\begin{array}{r}\mathbb{R}\text{(real-valued)}\\ \mathbb{Z}\text{(integer-valued)}\end{array}$$

Example

$$\begin{array}{rlrlr}A=\{a,b,c\}& & & & \\ C=\{1,2,3\}& & & & \\ & & & & \\ g:A\to A& & & & \\ g(a)=b& & g(b)=c& & g(c)=a\\ & & & & \\ f:A\to C& & & & \\ f(a)=3& & f(b)=2& & f(c)=1\\ & & & & \\ f\circ g:A\to C& & & & \\ f\circ g(a)=f(g(a))& & & & \\ =f(b)& & & & \\ =2& & & & \\ & & & & \\ f\circ g(b)=f(g(b))& & & & \\ =f(c)& & & & \\ =1& & & & \\ & & & & \\ f(g(c))=f(a)& & & & \\ =3& & & & \end{array}\text{NOTE:}g\circ f\text{NOT DEFINED BECAUSE g IS A SUBSET OF f}$$