GCSEFurther MathsAQARevision Notes2. Algebra & FunctionsFunctionsComposite & Inverse Functions

Composite & Inverse Functions (AQA GCSE Further Maths)

Revision Note

Test yourself
Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Did this video help you?

Composite Functions

What is a composite function?

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

  • Composite functions may also be referred to as compound functions

What do composite functions look like?

  • The notation you will see for a composite function is fg(x)

    • This can be written as f(g(x)) and means “f applied to the output of g(x)” 

    • i.e. g(x) happens first

  • Always apply the function on the outside to the output of the function on the inside

    • gf(x) means g(f(x)) and means “g applied to the output of f(x)” 

    • i.e. f(x) happens first

How does a composite function work?

  • If you are putting a number into fg(x)

    • STEP 1
      Put the number into g(x)

    • STEP 2
      Put the output of g(x) into f(x)

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

      • fg left parenthesis 2 right parenthesis space equals space straight f left parenthesis 1 half right parenthesis space equals 2 space cross times space 1 half space plus space 1 space equals space 2

      • gf left parenthesis 2 right parenthesis space equals space straight g left parenthesis 2 space cross times space 2 space plus space 1 right parenthesis space equals space straight g left parenthesis 5 right parenthesis space equals space 1 fifth

  • If you are using algebra, to find an expression for a composite function

    • STEP 1
      For fg(x) put g(x) wherever you see x in f(x)

    • STEP 2
      Simplify if necessary

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

      • fg left parenthesis x right parenthesis space equals space straight f left parenthesis 1 over x right parenthesis space equals space 2 space cross times space 1 over x space plus space 1 space equals space fraction numerator 2 space over denominator x end fraction plus space 1

      • gf left parenthesis x right parenthesis space equals space straight g left parenthesis 2 x space plus space 1 right parenthesis space equals space fraction numerator 1 over denominator 2 x space plus space 1 end fraction

Examiner Tips and Tricks

  • Make sure you are applying the functions in the correct order

    • The letter nearest the bracket is the function applied first

Worked Example

In this question, straight f open parentheses x close parentheses space equals space 2 x space minus space 1 and straight g open parentheses x close parentheses space equals space 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. 

fg stretchy left parenthesis 4 stretchy right parenthesis space equals space straight f open parentheses straight g open parentheses 4 close parentheses close parentheses space equals space straight f open parentheses open parentheses 4 space plus space 2 close parentheses squared close parentheses space equals space straight f open parentheses 6 squared close parentheses space equals space straight f open parentheses 36 close parentheses 

Apply f to the output of g.

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

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

(b) Find  g straight f open parentheses x close parentheses.
 
f is on the inside of the composite function so apply f first by substituting the function f(x) into g(x).

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

Simplify

table row cell g straight f open parentheses x close parentheses space end cell equals cell space open parentheses 2 x space minus space 1 space plus space 2 close parentheses squared end cell end table  

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

Did this video help you?

Inverse Functions

What is an inverse function?

  • An inverse function does the exact opposite of the function it came from

    • For example, if the function “doubles the number and adds 1” then its inverse is

    • “subtract 1 and halve the result”

  • It is the inverse operations in the reverse order

How do I write inverse functions?

  • An inverse function f-1 can be written as space straight f to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space horizontal ellipsis space space 

    • For example, if straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1 its inverse can be written as

    • straight f to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space fraction numerator left parenthesis x space – space 1 right parenthesis space over denominator 2 end fraction  

How do I find an inverse function?

  • The easiest way to find an inverse function is to 'cheat' and swap the x and y variables

    • Note that this is a useful method but you MUST remember not to do this in any other circumstances in maths

    • STEP 1
      Write the function in the formspace y space equals space horizontal ellipsis

    • STEP 2
      Swap the xs and space ys to get x space equals space horizontal ellipsis

    • STEP 3
      Rearrange the expression to make y the subject again

    • STEP 4
      Write as f-1(x) = … 

      • y should not exist in the final answer

  • For example, if straight f left parenthesis x right parenthesis equals 2 x plus 1 its inverse can be found as follows 

    • STEP 1
      Write the function in the form y space equals space 2 x space plus space 1

    • STEP 2
      Swap the x and y to get x space equals space 2 y space plus space 1

    • STEP 3
      Rearrange the expression to make y the subject again

table row cell x space minus space 1 space end cell equals cell space 2 y end cell row cell fraction numerator x space minus space 1 over denominator 2 end fraction space end cell equals cell space y space space space space space space rightwards arrow space space space y space equals space fraction numerator x space minus space 1 over denominator 2 end fraction end cell end table

  • STEP 4
    Rewrite using the correct notation for an inverse function

    • straight f to the power of negative 1 end exponent open parentheses x close parentheses space equals space fraction numerator space x italic space minus space 1 over denominator 2 end fraction

How does a function relate to its inverse?

  • If straight f open parentheses 3 close parentheses equals 10 then the input of 3 gives an output of 10

    • The inverse function undoes f(x)

    • An input of 10 into the inverse function gives an output of 3

      • If f open parentheses 3 close parentheses equals 10 then straight f to the power of negative 1 end exponent open parentheses 10 close parentheses equals 3

  • ff to the power of negative 1 end exponent open parentheses x close parentheses equals straight f to the power of negative 1 end exponent straight f open parentheses x close parentheses equals x

    • If you apply a function to x, then immediately apply its inverse function, you get x

      • Whatever happened to x gets undone

    • f and f-1 cancel each other out when applied together

  • If straight f open parentheses x close parentheses space equals space 2 to the power of x and you want to solve straight f to the power of negative 1 end exponent open parentheses x close parentheses space equals space 5

    • Finding the inverse function straight f to the power of negative 1 end exponent open parentheses x close parentheses in this case is tricky (impossible if you haven't studied logarithms)

    • instead, take f of both sides and use that ff to the power of negative 1 end exponent cancel each other out:

table attributes columnalign right center left columnspacing 0px end attributes row cell ff to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell straight f open parentheses 5 close parentheses end cell row x equals cell straight f open parentheses 5 close parentheses end cell row x equals cell 2 to the power of 5 equals 32 end cell end table

Worked Example

Find the inverse of the function straight f open parentheses x close parentheses space equals space 5 space minus space 3 x.
 
Write the function in the form y space equals space 5 space minus space 3 x and then swap the x and y.

y space equals space 5 space minus space 3 x x space equals space 5 space minus space 3 y  

Rearrange the expression to make y the subject again.

table row cell x space end cell equals cell space 5 space minus space 3 y end cell row cell space x space plus space 3 y space end cell equals cell space 5 end cell row cell 3 y space end cell equals cell space 5 space minus space x end cell row cell y space end cell equals cell space fraction numerator 5 space minus space x over denominator 3 end fraction end cell end table

Rewrite using the correct notation for an inverse function.

Last updated:

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:Functions ToolkitNext:Finding nth Terms of Sequences
    Author
    Reviewer
    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.