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Composite Functions (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

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Composite Functions

What is a composite function?

  • A composite function is where a function is applied to another function

  • A composite function can be denoted

    • left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis

    • space f g left parenthesis x right parenthesis

    • space f stretchy left parenthesis g left parenthesis x stretchy right parenthesis right parenthesis

  • The order matters

    • left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis means:

      • First apply g to x to get space g left parenthesis x right parenthesis

      • Then apply f to the previous output to get space f stretchy left parenthesis g left parenthesis x stretchy right parenthesis right parenthesis

      • Always start with the function closest to the variable

    • left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis is not usually equal to left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis

How do I find the domain and range of a composite function?

  • The domain of space f ring operator g is the set of values of x...

    • which are a subset of the domain of g

    • which maps g to a value that is in the domain of f

  • The range of space f ring operator g is the set of values of x...

    • which are a subset of the range of f

    • found by applying f to the range of g

  • To find the domain and range of space f ring operator g

    • First find the range of g

    • Restrict these values to the values that are within the domain of f

      • The domain is the set of values that produce the restricted range of g

      • The range is the set of values that are produced using the restricted range of g as the domain for f

  • For example: let space f left parenthesis x right parenthesis equals 2 x plus 1 comma space minus 5 less or equal than x less or equal than 5 and space g left parenthesis x right parenthesis equals square root of x comma space 1 less or equal than x less or equal than 49

    • The range of g is 1 less or equal than g left parenthesis x right parenthesis less or equal than 7

      • Restricting this to fit the domain of f results in 1 less or equal than g left parenthesis x right parenthesis less or equal than 5

    • The domain of space f ring operator g is therefore 1 less or equal than x less or equal than 25

      • These are the values of x which map to 1 less or equal than g left parenthesis x right parenthesis less or equal than 5

    • The range of space f ring operator g is therefore 3 less or equal than left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis less or equal than 11

      • These are the values which f maps 1 less or equal than g left parenthesis x right parenthesis less or equal than 5 to

Examiner Tips and Tricks

  • Make sure you know what your GDC is capable of with regard to functions

    • You may be able to store individual functions and find composite functions and their values for particular inputs

    • You may be able to graph composite functions directly and so deduce their domain and range from the graph

  • The link between the domains and ranges of a function and its inverse can act as a check for your solution

  • space f f left parenthesis x right parenthesis is not the same as stretchy left square bracket f left parenthesis x right parenthesis stretchy right square bracket squared

Worked Example

Given space f left parenthesis x right parenthesis equals square root of x plus 4 end root and space g left parenthesis x right parenthesis equals 3 plus 2 x:

a) Write down the value of left parenthesis g ring operator f right parenthesis left parenthesis 12 right parenthesis.

2-3-2-ib-aa-sl-composite-functions-a-we-solution

b) Write down an expression for left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis.

2-3-2-ib-aa-sl-composite-functions-b-we-solution

c) Write down an expression for left parenthesis g ring operator g right parenthesis left parenthesis x right parenthesis.

2-3-2-ib-aa-sl-composite-functions-c-we-solution

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    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject 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.