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Composite & Inverse Functions (Cambridge O Level Maths)

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Composite Functions

What is a composite function?

  • A composite function is one function applied to the output of another function
  • Composite functions may also 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 Tip

  • 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 bold 2

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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 or  straight f to the power of negative 1 end exponent space colon space x space rightwards arrow from bar space horizontal ellipsis
    • 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  or   straight f to the power of negative 1 end exponent colon space x space rightwards arrow from bar space fraction numerator left parenthesis x space – space 1 right parenthesis 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) = … (or 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

How do I find the domain and range of an inverse function?

  • The domain of an inverse function is the same as the range of the original function
  • The range of an inverse function is the same as the domain of the original function

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 attributes columnalign right center left columnspacing 0px end attributes 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

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.