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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 2. Functions / 2.4 Functions Toolkit / 2.4.1 Composite & Inverse Functions


2.4.1 Composite & Inverse Functions


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

Exam Tip

  • 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

Inverse Functions

What is an inverse function?

  • Only one-to-one functions have inverses
  • A function has an inverse if its graph passes the horizontal line test
    • Any horizontal line will intersect with the graph at most once
  • The identity function id maps each value to itself
    • id left parenthesis x right parenthesis equals x
  • If space f ring operator g and space g ring operator f have the same effect as the identity function then space f and space g are inverses
  • Given a function space f left parenthesis x right parenthesis we denote the inverse function as space f to the power of negative 1 end exponent left parenthesis x right parenthesis
  • An inverse function reverses the effect of a function
    • space f left parenthesis 2 right parenthesis equals 5 means space f to the power of negative 1 end exponent left parenthesis 5 right parenthesis equals 2
  • Inverse functions are used to solve equations
    • The solution of space f left parenthesis x right parenthesis equals 5 is x equals f to the power of negative 1 end exponent left parenthesis 5 right parenthesis
  • A composite function made of space f and space f to the power of negative 1 end exponent has the same effect as the identity function
    • left parenthesis f ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals left parenthesis f to the power of negative 1 end exponent ring operator f right parenthesis left parenthesis x right parenthesis equals x

Language of Functions Notes Diagram 9

What are the connections between a function and its inverse function?

  • The domain of a function becomes the range of its inverse
  • The range of a function becomes the domain of its inverse
  • The graph of space y equals f to the power of negative 1 end exponent left parenthesis x right parenthesis is a reflection of the graph space y equals f left parenthesis x right parenthesis in the line space y equals x
    • Therefore solutions to space f left parenthesis x right parenthesis equals x or space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals x will also be solutions to space f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis x right parenthesis
      • There could be other solutions to space f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis x right parenthesis that don't lie on the line space y equals x

Inverse Functions Notes Diagram 2

How do I find the inverse of a function?

  • STEP 1: Swap the x and y in space y equals f left parenthesis x right parenthesis
    • If space y equals f to the power of negative 1 end exponent left parenthesis x right parenthesis then x equals f left parenthesis y right parenthesis
  • STEP 2: Rearrange x equals f left parenthesis y right parenthesis to make space y the subject
  • Note this can be done in any order
    • Rearrange space y equals f left parenthesis x right parenthesis to make x the subject
    • Swap x and space y

Can many-to-one functions ever have inverses?

  • You can restrict the domain of a many-to-one function so that it has an inverse
  • Choose a subset of the domain where the function is one-to-one
    • The inverse will be determined by the restricted domain
    • Note that a many-to-one function can only have an inverse if its domain is restricted first
  • For quadratics – use the vertex as the upper or lower bound for the restricted domain
    • For space f left parenthesis x right parenthesis equals x squared restrict the domain so 0 is either the maximum or minimum value
      • For example: x greater or equal than 0 or x less or equal than 0
    • For space f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k restrict the domain so h is either the maximum or minimum value
      • For example: x greater or equal than h or x less or equal than h
  • For trigonometric functions – use part of a cycle as the restricted domain
    • For space f left parenthesis x right parenthesis equals sin x restrict the domain to half a cycle between a maximum and a minimum
      • For example: negative pi over 2 less or equal than x less or equal than pi over 2
    • For space f left parenthesis x right parenthesis equals cos x restrict the domain to half a cycle between maximum and a minimum
      • For example: 0 less or equal than x less or equal than pi
    • For space f left parenthesis x right parenthesis equals tan x restrict the domain to one cycle between two asymptotes
      • For example: negative pi over 2 less than x less than pi over 2

How do I find the inverse function after restricting the domain?

  • The range of the inverse is the same as the restricted domain of the original function
  • The inverse function is determined by the restricted domain
    • Restricting the domain differently will change the inverse function
  • Use the range of the inverse to help find the inverse function
    • Restricting the domain of space f left parenthesis x right parenthesis equals x squared to x greater or equal than 0 means the range of the inverse is space f to the power of negative 1 end exponent left parenthesis x right parenthesis greater or equal than 0
      • Therefore space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals square root of x
    • Restricting the domain of space f left parenthesis x right parenthesis equals x squared to x less or equal than 0 means the range of the inverse is space f to the power of negative 1 end exponent left parenthesis x right parenthesis less or equal than 0
      • Therefore space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals negative square root of x

Exam Tip

  • Remember that an inverse function is a reflection of the original function in the line y equals x
    • Use your GDC to plot the function and its inverse on the same graph to visually check this
  • space f to the power of negative 1 end exponent left parenthesis x right parenthesis  is not the same as  fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction

Worked Example

The function space f open parentheses x close parentheses equals open parentheses x minus 2 close parentheses squared plus 5 comma blank x less or equal than m has an inverse.

a)
Write down the largest possible value of m.

d7t4IIb~_2-3-2-ib-aa-hl-inverse-functions-a-we-solution

b)
Find the inverse of space f left parenthesis x right parenthesis.

2-3-2-ib-aa-hl-inverse-functions-b-we-solution

c)
Find the domain of space f to the power of negative 1 end exponent left parenthesis x right parenthesis.

2-3-2-ib-aa-hl-inverse-functions-c-we-solution

d)
Find the value of k such that space f left parenthesis k right parenthesis equals 9.

2-3-2-ib-aa-hl-inverse-functions-d-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
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              • 1.2.2 Logarithms
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse Functions
                                                                        • 2.5 Transformations of Graphs
                                                                          • 2.5.1 Translations of Graphs
                                                                            • 2.5.2 Reflections of Graphs
                                                                              • 2.5.3 Stretches of Graphs
                                                                                • 2.5.4 Composite Transformations of Graphs
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 3. Geometry & Trigonometry
                                                                                        • 3.1 Geometry Toolkit
                                                                                          • 3.1.1 Coordinate Geometry
                                                                                            • 3.1.2 Radian Measure
                                                                                              • 3.1.3 Arcs & Sectors
                                                                                              • 3.2 Geometry of 3D Shapes
                                                                                                • 3.2.1 3D Coordinate Geometry
                                                                                                  • 3.2.2 Volume & Surface Area
                                                                                                  • 3.3 Trigonometry
                                                                                                    • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                      • 3.3.2 Non Right-Angled Trigonometry
                                                                                                        • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                        • 3.4 Further Trigonometry
                                                                                                          • 3.4.1 The Unit Circle
                                                                                                            • 3.4.2 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      Daniel Finlay

                                                                                                                                                                                                                                                                                      Author: Daniel

                                                                                                                                                                                                                                                                                      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.


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