Composite Functions

Composite functions

1.2

Composite Functions

This image is an educational diagram explaining composite functions, denoted as gf(x). It is divided into four sections: 1. ‘Definition’: A function that maps other functions. 2. ‘Notation’: Includes g(f(x)), f(g(x)), and g∘f(x). 3. ‘Domain’: The first function, usually f(x) or x. 4. ‘Range’: The composite function itself, either gf(x) or fg(x). A pencil illustration is in the center, pointing to the term ‘Composite Function gf(x)’ with arrows connecting to each section.

Example:

Visual representation of composite function with three sets P, Q, and R, and functions f and g mapping between them.

Function \(f\) maps set \(P\) to set \(Q\), function \(g\) maps set \(Q\) to set \(R\) and function \(gf\) maps set \(P\) to set \(R\).
Given two functions \(f(x)\) and \(g(x)\), both functions can be combined and written as \(fg(x)\) or \(gf(x)\) which is defined as \(fg(x) = f[g(x)]\) or \(gf(x)=g[f(x)]\).
In general, \(fg \neq gf\), \(f^2=ff\), \(f^3=fff\), and so on.

Composite functions

1.2

Composite Functions

This image is an educational diagram explaining composite functions, denoted as gf(x). It is divided into four sections: 1. ‘Definition’: A function that maps other functions. 2. ‘Notation’: Includes g(f(x)), f(g(x)), and g∘f(x). 3. ‘Domain’: The first function, usually f(x) or x. 4. ‘Range’: The composite function itself, either gf(x) or fg(x). A pencil illustration is in the center, pointing to the term ‘Composite Function gf(x)’ with arrows connecting to each section.

Example:

Visual representation of composite function with three sets P, Q, and R, and functions f and g mapping between them.

Function \(f\) maps set \(P\) to set \(Q\), function \(g\) maps set \(Q\) to set \(R\) and function \(gf\) maps set \(P\) to set \(R\).
Given two functions \(f(x)\) and \(g(x)\), both functions can be combined and written as \(fg(x)\) or \(gf(x)\) which is defined as \(fg(x) = f[g(x)]\) or \(gf(x)=g[f(x)]\).
In general, \(fg \neq gf\), \(f^2=ff\), \(f^3=fff\), and so on.