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Composite functions

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Summary

Composite functions

​​In a nutshell

Composite functions consist of two or more functions combined. The output of one function is used as the input of another function. The order of the functions matters. fg(x)fg(x)fg(x) means to apply ggg first, then fff.​


​f(g(x))=fg(x)≠gf(x)\boxed{f(g(x))=fg(x)\not=gf(x)}f(g(x))=fg(x)=gf(x)​​​


Composite functions

A composite function takes an input xxx and applies two functions to the input, one after another. In this way, two or more functions can be chained to create more complex ones. The output result of the composite function depends on the order of the composition.


When given two functions like f(x)=2xf(x)=2xf(x)=2x​ and g(x)=x2g(x)=x^2g(x)=x2, there are two possible compositions:

  • f(g(x))=f(x2)=2x2f(g(x))=f(x^2)=2x^2f(g(x))=f(x2)=2x2​​
  • ​g(f(x))=g(2x)=(2x)2=4x2g(f(x))=g(2x)=(2x)^2=4x^2g(f(x))=g(2x)=(2x)2=4x2​​


Examples

Given f(x)=x2+1f(x)=x^2+1f(x)=x2+1 and g(x)=2x+2g(x)=2x+2g(x)=2x+2, find:

a) fg(x)fg(x)fg(x)

b) gf(x)gf(x)gf(x)

c) ff(x)ff(x)ff(x)​

​​

a) Find fg(x)fg(x)fg(x) by substituting ggg into fff:

fg(x)=f(2x+2)=(2x+2)2+1=4x2+4x+4+1=4x2+4x+5‾\begin{aligned}fg(x)&=f(2x+2) \\& =(2x+2)^2+1\\&=4x^2+4x+4+1\\&=\underline{4x^2+4x+5}\end{aligned}fg(x)​=f(2x+2)=(2x+2)2+1=4x2+4x+4+1=4x2+4x+5​​​​


b) Substitute fff into ggg:

​gf(x)=g(x2+1)=2(x2+1)+2=2x2+2+2=2x2+4‾\begin{aligned}gf(x)&=g(x^2+1) \\&=2(x^2+1)+2\\&=2x^2+2+2\\&=\underline{2x^2+4}\end{aligned}gf(x)​=g(x2+1)=2(x2+1)+2=2x2+2+2=2x2+4​​​​


c) Substitute fff into fff:

​ff(x)=f(x2+1)=(x2+1)2+1=x4+2x2+1+1=x4+2x2+2‾\begin{aligned}ff(x)&=f(x^2+1) \\&=(x^2+1)^2+1\\&=x^4+2x^2+1+1\\&=\underline{x^4+2x^2+2}\end{aligned}ff(x)​=f(x2+1)=(x2+1)2+1=x4+2x2+1+1=x4+2x2+2​​​​


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Expressions, equations, formulae, functions and identities

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Expressions, equations, formulae, functions and identities

Composite and inverse functions - Higher

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Composite and inverse functions - Higher

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Composite functions

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Composite functions

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FAQs - Frequently Asked Questions

How do I read a composite function?

The composite function gf(x) is read as "g of f of x".

Does the order of a composite function alter the result?

Yes, in most cases the order of the composite function matters. fg(x) is different to gf(x)

What is a composite function?

A function made up of two or more functions.

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