Composite Functions (Edexcel IGCSE Maths A (Modular))

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Composite Functions

What is a composite function?

  • A composite function is a function applied to the output of another function

    • Composite functions may also referred to as compound functions

What notation is used for composite functions?

  • If straight f open parentheses x close parentheses and straight g open parentheses x close parentheses are two functions, then

    • straight g open parentheses straight f open parentheses x close parentheses close parentheses is a composite function

      • Also written gf colon x

    • It means the input xgoes through function straight f first

      • This gives the output straight f open parentheses x close parentheses

      • Then this output, straight f open parentheses x close parentheses, becomes the input of function straight g, giving straight g open parentheses straight f open parentheses x close parentheses close parentheses

    • gf open parentheses x close parentheses is the shorthand notation used for straight g open parentheses straight f open parentheses x close parentheses close parentheses

      • It means do straight f first, then straight g

      • The order of applying the functions goes from right to left

      • (the letter nearest the bracket goes first)

      • This is often the opposite of what people expect!

    • fg open parentheses x close parentheses means do straight g open parentheses x close parentheses first then straight f open parentheses x close parentheses second

    • ff open parentheses x close parentheses means apply straight f open parentheses x close parentheses twice!

      • This can be written straight f squared open parentheses x close parentheses

      • This does not mean the same as open square brackets straight f open parentheses x close parentheses close square brackets squared

Examiner Tips and Tricks

A good trick in the exam is to write brackets around gf open parentheses x close parentheses to make it straight g open parentheses straight f open parentheses x close parentheses close parentheses, to see that it is "g" of "f(x)".

How do I substitute numbers into composite functions?

  • If you are putting a number into a composite function

    • put the number into the function closest to (x)

    • then make the output of the first function the input of the second function

  • For example, if straight f open parentheses x close parentheses equals 2 x plus 1 and straight g open parentheses x close parentheses equals 1 over x

    • to find gf open parentheses 2 close parentheses:

      • Put the 2 in as the input of straight f first

      • straight f left parenthesis 2 right parenthesis equals 2 open parentheses 2 close parentheses plus 1 equals 5 space

      • Then put 5 in as the input of straight g

      • So gf open parentheses 2 close parentheses equals straight g open parentheses straight f open parentheses 2 close parentheses close parentheses equals straight g open parentheses 5 close parentheses equals 1 fifth

    • to find fg open parentheses 2 close parentheses:

      • Put the 2 in as the input of straight g first

      • straight g open parentheses 2 close parentheses equals 1 half

      • Then put 1 half in as the input of straight f

      • So fg open parentheses 2 close parentheses equals straight f open parentheses straight g open parentheses 2 close parentheses close parentheses equals straight f open parentheses 1 half close parentheses equals 2 open parentheses 1 half close parentheses plus 1 equals 2

    • to find ff open parentheses 2 close parentheses:

      • straight f open parentheses 2 close parentheses equals 2 cross times 2 plus 1 equals 5

      • straight f open parentheses 5 close parentheses equals 2 cross times 5 plus 1 equals 11

      • so ff open parentheses 2 close parentheses equals 11

How do I find composite functions algebraically?

  • If you are using algebra, substitute the whole algebraic expression as your input

    • For example, if straight f left parenthesis x right parenthesis equals 2 x plus 1 and straight g left parenthesis x right parenthesis equals 1 over x

      • fg left parenthesis x right parenthesis equals straight f open parentheses straight g open parentheses x close parentheses close parentheses equals straight f open parentheses 1 over x close parentheses equals 2 cross times open parentheses 1 over x close parentheses plus 1 equals 2 over x plus 1

      • gf left parenthesis x right parenthesis equals straight g open parentheses straight f open parentheses x close parentheses close parentheses equals straight g left parenthesis 2 x plus 1 right parenthesis equals fraction numerator 1 over denominator 2 x plus 1 end fraction

      • ff open parentheses x close parentheses equals straight f open parentheses straight f open parentheses x close parentheses close parentheses equals straight f open parentheses 2 x plus 1 close parentheses equals 2 open parentheses 2 x plus 1 close parentheses plus 1 which simplifies to ff open parentheses x close parentheses equals 4 x plus 3

Worked Example

In this question, straight f open parentheses x close parentheses space equals space 2 x space minus space 1 and straight g colon x rightwards arrow open parentheses x space plus space 2 close parentheses squared.

(a) Find  fg open parentheses 4 close parentheses.

"g" is on the inside of the composite function, so apply g first

straight g open parentheses 4 close parentheses equals open parentheses 4 plus 2 close parentheses squared equals 6 squared equals 36

Now apply the function "f" to 36

table row cell space straight f open parentheses 36 close parentheses end cell equals cell 2 open parentheses 36 close parentheses minus 1 end cell row blank equals cell 72 minus 1 end cell end table

bold fg bold left parenthesis bold 4 bold right parenthesis bold equals bold 71

(b) Find  gf open parentheses x close parentheses.

"f" is on the inside of the composite function so substitute the function f(x) into g(x)
It can help to write gf open parentheses x close parentheses equals straight g open parentheses straight f open parentheses x close parentheses close parentheses

 gf stretchy left parenthesis x stretchy right parenthesis equals straight g open parentheses straight f open parentheses x close parentheses close parentheses space equals straight g open parentheses 2 x minus 1 close parentheses equals open parentheses open parentheses 2 x minus 1 close parentheses plus 2 close parentheses squared

Simplify inside the bracket

table row cell gf open parentheses x close parentheses end cell equals cell open parentheses 2 x minus 1 plus 2 close parentheses squared end cell end table

bold gf bold left parenthesis bold italic x bold right parenthesis bold equals bold left parenthesis bold 2 bold italic x bold plus bold 1 bold right parenthesis to the power of bold 2

You do not need to expand the answer

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Mark Curtis

Author: Mark Curtis

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.