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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.
 
Definition
Given two functions \(f(x)\) and \(g(x)\), the product of combination of two functions that written as \(fg(x)\) or \(gf(x)\) are defined by \(fg(x)=f[g(x)]\) or \(gf(x)=g[f(x)]\).
 
Properties of Composite Function
Figure

Visual representation of composite function with three sets P, Q, and R, and functions f and g mapping between them.
Description
  • 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.
 
Example \(1\)
Question

Given two functions \(f(x)=2x\) and \(g(x)=x^2-5\).

Determine the following composite functions.

(a) \(fg\)
(b) \(gf\)
(c) \(f^2\)
(d) \(g^2\)
Solution

(a)

\(\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-5) \\ &=2(x^2-5) \\ &=2x^2-10. \end{aligned}\)

\(\therefore fg(x)=2x^2-10.\)

(b)

\(\begin{aligned} gf(x)&=g[f(x)] \\ &=g(2x) \\ &=(2x)^2-5 \\ &=4x^2-5. \end{aligned}\)

\(\therefore gf(x)=4x^2-5.\)

(c)

\(\begin{aligned} f^2(x)&=f[f(x)] \\ &=f(2x) \\ &=2(2x) \\ &=4x .\end{aligned}\)

\(\therefore f^2(x)=4x.\)

(d)

\(\begin{aligned} g^2(x)&=g[g(x)] \\ &=g(x^2-5) \\ &=(x^2-5)^2-5 \\ &=x^4-10x^2+25-5\\ &=x^4-10x^2+20. \end{aligned}\)

\(\therefore g^2(x)=x^4-10x^2+20.\)
 
Example \(2\)
Question

If \(f(x)=x-1\) and \(g(x)=x^2-3x+4\), find

(a) \(fg(2)\),
(b) the values of \(x\) when \(fg(x)=7\).
Solution

(a)

\(\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-3x+4) \\ &=x^2-3x+4-1 \\ &=x^2-3x+3. \end{aligned}\)

Thus,

\(\begin{aligned} fg(2)&=(2)^2-3(2)+3 \\ &=1. \end{aligned}\)

(b)

\(\begin{aligned} fg(x)&=7 \\ x^2-3x+3&=7 \\ x^2-3x-4&=0 \\ (x+1)(x-4)&=0. \end{aligned}\)

Thus,

\(\therefore x=-1,\quad x=4.\)
 
Example \(3\)
Question

Given function \(f(x)=x-2\). Find the function \(g(x)\) if \(fg(x)=8x-7\).
Solution

\(\begin{aligned} f[g(x)]&=8x-7 \\ g(x)-2&=8x-7 \\ g(x)&=8x-7+2. \\ \end{aligned}\)

\(\therefore g(x)=8x-5.\)
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.

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.
 
Definition
Given two functions \(f(x)\) and \(g(x)\), the product of combination of two functions that written as \(fg(x)\) or \(gf(x)\) are defined by \(fg(x)=f[g(x)]\) or \(gf(x)=g[f(x)]\).
 
Properties of Composite Function
Figure

Visual representation of composite function with three sets P, Q, and R, and functions f and g mapping between them.
Description
  • 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.
 
Example \(1\)
Question

Given two functions \(f(x)=2x\) and \(g(x)=x^2-5\).

Determine the following composite functions.

(a) \(fg\)
(b) \(gf\)
(c) \(f^2\)
(d) \(g^2\)
Solution

(a)

\(\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-5) \\ &=2(x^2-5) \\ &=2x^2-10. \end{aligned}\)

\(\therefore fg(x)=2x^2-10.\)

(b)

\(\begin{aligned} gf(x)&=g[f(x)] \\ &=g(2x) \\ &=(2x)^2-5 \\ &=4x^2-5. \end{aligned}\)

\(\therefore gf(x)=4x^2-5.\)

(c)

\(\begin{aligned} f^2(x)&=f[f(x)] \\ &=f(2x) \\ &=2(2x) \\ &=4x .\end{aligned}\)

\(\therefore f^2(x)=4x.\)

(d)

\(\begin{aligned} g^2(x)&=g[g(x)] \\ &=g(x^2-5) \\ &=(x^2-5)^2-5 \\ &=x^4-10x^2+25-5\\ &=x^4-10x^2+20. \end{aligned}\)

\(\therefore g^2(x)=x^4-10x^2+20.\)
 
Example \(2\)
Question

If \(f(x)=x-1\) and \(g(x)=x^2-3x+4\), find

(a) \(fg(2)\),
(b) the values of \(x\) when \(fg(x)=7\).
Solution

(a)

\(\begin{aligned} fg(x)&=f[g(x)] \\ &=f(x^2-3x+4) \\ &=x^2-3x+4-1 \\ &=x^2-3x+3. \end{aligned}\)

Thus,

\(\begin{aligned} fg(2)&=(2)^2-3(2)+3 \\ &=1. \end{aligned}\)

(b)

\(\begin{aligned} fg(x)&=7 \\ x^2-3x+3&=7 \\ x^2-3x-4&=0 \\ (x+1)(x-4)&=0. \end{aligned}\)

Thus,

\(\therefore x=-1,\quad x=4.\)
 
Example \(3\)
Question

Given function \(f(x)=x-2\). Find the function \(g(x)\) if \(fg(x)=8x-7\).
Solution

\(\begin{aligned} f[g(x)]&=8x-7 \\ g(x)-2&=8x-7 \\ g(x)&=8x-7+2. \\ \end{aligned}\)

\(\therefore g(x)=8x-5.\)
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.
Chapter : Functions
Topic : Solve problems involving composite functions
Form 4 Additional Mathematics
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