Linear Algebra - Function composition

Card Puncher Data Processing


Functions + Class - Composition (Composite | Component)

  • <math>f : A \rightarrow B</math>
  • and <math>g : B \rightarrow C</math>

the functional composition of f and g is the function:

  • <math>(g \circ f) : A rightarrow C</math>

defined by:

  • <math>(g \circ f)(x) = g(f (x))</math>


Example <math>g \circ f</math>
<math>f : \lbrace {1, 2, 3} \rbrace \rightarrow \lbrace { A, B, C, D } \rbrace</math>
<math>g : \lbrace{A, B, C, D}\rbrace \rightarrow \lbrace{4, 5}\rbrace</math>
Function Composition
<math>g(y) = y%%^%%2</math>
<math>f(x) = x + 1 </math>
<math>(g \circ f)(x) = (x + 1)%%^%%2</math>
<math>f: \lbrace {A, B, C, ..., Z} \rbrace \rightarrow \lbrace {0, 1, 2, ..., 25}\rbrace</math>
<math>g(x) = (x+3) mod 26</math>
<math>h: \lbrace{A, B, C, ..., Z}\rbrace \rightarrow \lbrace{0, 1, 2, ..., 25}\rbrace</math>
<math>h \circ (g \circ f)</math> is the Caesar cypher
Fogoh Caesar Fogoh Caesar Simplify


Associativity of function composition: <math>h \circ (g \circ f) = (h \circ g) \circ f</math>

<math>(h \circ (g \circ f))(x)</math> =

  • <math>h(g \circ f)(x)</math>
  • <math>h(g(f(x)))</math>
  • <math>(h \circ g)(f(x))</math>
  • <math>((h \circ g) \circ f)(x)</math>

Discover More
Card Puncher Data Processing
Automata - Deterministic finite-state automata (DFA)

A Deterministic finite-state automata (DFA) is a finite automaton that cannot be in more than one state at any one time. The term deterministic refers to the fact that on each input there is one and only...
Ggplot Graphic Plot
Ggplot - History

ggplot worked using function composition instead of addition. So instead of addition: You wrote something a composition: An interesting historical...
Function Domain Codomain Caesar Crypto
Linear Algebra - Function (Set)

For each input element in a set , a function assigns a single output element from another set . is called the domain of the function is called the co-domain or in Mathese: The function...

Share this page:
Follow us:
Task Runner