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DP IB Maths: AI SL

Revision Notes

Home / IB / Maths: AI SL / DP / Revision Notes / 2. Functions / 2.2 Further Functions & Graphs / 2.2.1 Functions


2.2.1 Functions


Language of Functions

What is a mapping?

  • A mapping transforms one set of values (inputs) into another set of values (outputs)
  • Mappings can be:
    • One-to-one
      • Each input gets mapped to exactly one unique output
      • No two inputs are mapped to the same output
      • For example: A mapping that cubes the input
    • Many-to-one
      • Each input gets mapped to exactly one output
      • Multiple inputs can be mapped to the same output
      • For example: A mapping that squares the input
    • One-to-many
      • An input can be mapped to more than one output
      • No two inputs are mapped to the same output
      • For example: A mapping that gives the numbers which when squared equal the input
    • Many-to-many
      • An input can be mapped to more than one output
      • Multiple inputs can be mapped to the same output
      • For example: A mapping that gives the factors of the input

Language of Functions Notes Diagram 2

What is a function?

  • A function is a mapping between two sets of numbers where each input gets mapped to exactly one output
    • The output does not need to be unique

  • One-to-one and many-to-one mappings are functions
  • A mapping is a function if its graph passes the vertical line test
    • Any vertical line will intersect with the graph at most once

Language of Functions Notes Diagram 4

What notation is used for functions?

  • Functions are denoted using letters (such as space f comma space v comma space g comma etc)
    • A function is followed by a variable in a bracket
    • This shows the input for the function
    • The letterspace f is used most commonly for functions and will be used for the remainder of this revision note
  • space f left parenthesis x right parenthesis represents an expression for the value of the function space f when evaluated for the variable x
  • Function notation gets rid of the need for words which makes it universal
    • space f equals 5 whenspace x equals 2 can simply be written as space f left parenthesis 2 right parenthesis equals 5

What are the domain and range of a function?

  • The domain of a function is the set of values that are used as inputs
  • A domain should be stated with a function
    • If a domain is not stated then it is assumed the domain is all the real values which would work as inputs for the function
    • Domains are expressed in terms of the input
      • space x less or equal than 2
  • The range of a function is the set of values that are given as outputs
    • The range depends on the domain
    • Ranges are expressed in terms of the output
      • space f stretchy left parenthesis x right parenthesis greater or equal than 0
  • To graph a function we use the inputs as the x-coordinates and the outputs as the y-coordinates
    • space f left parenthesis 2 right parenthesis equals 5 corresponds to the coordinates (2, 5)
  • Graphing the function can help you visualise the range
  • Common sets of numbers have special symbols:
    • straight real numbers represents all the real numbers that can be placed on a number line
      • x element of straight real numbers means xis a real number
    • straight rational numbers represents all the rational numbers a over bwhere a and b are integers and b ≠ 0
    • straight integer numbers represents all the integers (positive, negative and zero)
      • straight integer numbers to the power of plus represents positive integers
    • straight natural numbers represents the natural numbers (0,1,2,3...)

2-3-1-sets-of-numbers-diagram

Exam Tip

  • Questions may refer to "the largest possible domain"
    • this would usually be  x element of straight real numbers  unless natural numbers, integers or quotients has already been stated
    • there are usually some exceptions
      • e.g.  square roots;  x greater or equal than 0  for a function involving  square root of x
      • e.g.  reciprocal functions;  x not equal to 2  for a function with denominator  left parenthesis x minus 2 right parenthesis  

Worked Example

For the function space f open parentheses x close parentheses equals x cubed plus 1 comma blank 2 less or equal than x less or equal than 10:

a)
write down the value of space f left parenthesis 7 right parenthesis.

2-2-1-ib-ai-sl-language-of-functions-a-we-solution

b)
find the range of space f left parenthesis x right parenthesis.

2-2-1-ib-ai-sl-language-of-functions-b-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
  • 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 space x equals f to the power of negative 1 end exponent left parenthesis 5 right parenthesis

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

Exam Tip

  • Remember that, in general,  f to the power of negative 1 end exponent left parenthesis x right parenthesis not equal to fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction

Worked Example

For the function space f open parentheses x close parentheses equals x cubed plus 1 comma blank 2 less or equal than x less or equal than 10:

a)
write down the range of the inverse function, space f to the power of negative 1 end exponent left parenthesis x right parenthesis.

2-2-1-ib-ai-sl-inverse-functions-a-we-solution-we-solution

b)
find the value of space f to the power of negative 1 end exponent left parenthesis 217 right parenthesis.

2-2-1-ib-ai-sl-inverse-functions-b-we-solution-we-solution

Piecewise Functions

What are piecewise functions?

  • Piecewise functions are defined by different functions depending on which interval the input is in
    • E.g. space f open parentheses x close parentheses equals open curly brackets table row cell x plus 1 end cell row cell 2 x minus 4 end cell end table blank table attributes columnalign left end attributes row cell x less or equal than 5 end cell row cell 5 less than x less than 10 end cell end table close
  • The region for the individual functions cannot overlap
  • To evaluate a piecewise function for a particular value space x equals k
    • Find which interval includes space k
    • Substitute space x equals k into the corresponding function

Worked Example

For the piecewise function

f open parentheses x close parentheses equals open curly brackets table row cell 2 x minus 5 end cell row cell 3 x plus 1 end cell end table blank table attributes columnalign left end attributes row cell negative 10 less or equal than x less or equal than 10 end cell row cell x greater than 10 end cell end table close,

a)
find the values of space f left parenthesis 0 right parenthesis comma space f left parenthesis 10 right parenthesis comma space f left parenthesis 20 right parenthesis.

2-2-1-ib-ai-sl-piecewise-functions-a-we-solution-we-solution

b)
state the domain.

2-2-1-ib-ai-sl-piecewise-functions-b-we-solution-we-solution



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Exponents & Logarithms
          • 1.1.3 Approximation & Estimation
            • 1.1.4 GDC: Solving Equations
            • 1.2 Sequences & Series
              • 1.2.1 Language of Sequences & Series
                • 1.2.2 Arithmetic Sequences & Series
                  • 1.2.3 Geometric Sequences & Series
                    • 1.2.4 Applications of Sequences & Series
                    • 1.3 Financial Applications
                      • 1.3.1 Compound Interest & Depreciation
                        • 1.3.2 Amortisation & Annuities
                      • 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 & Piecewise 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
                                          • 3. Geometry & Trigonometry
                                            • 3.1 Geometry Toolkit
                                              • 3.1.1 Coordinate Geometry
                                                • 3.1.2 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 Voronoi Diagrams
                                                            • 3.4.1 Voronoi Diagrams
                                                              • 3.4.2 Toxic Waste Dump Problem
                                                            • 4. Statistics & Probability
                                                              • 4.1 Statistics Toolkit
                                                                • 4.1.1 Sampling & Data Collection
                                                                  • 4.1.2 Statistical Measures
                                                                    • 4.1.3 Frequency Tables
                                                                      • 4.1.4 Linear Transformations of Data
                                                                        • 4.1.5 Outliers
                                                                          • 4.1.6 Univariate Data
                                                                            • 4.1.7 Interpreting Data
                                                                            • 4.2 Correlation & Regression
                                                                              • 4.2.1 Bivariate data
                                                                                • 4.2.2 Correlation Coefficients
                                                                                  • 4.2.3 Linear Regression
                                                                                  • 4.3 Probability
                                                                                    • 4.3.1 Probability & Types of Events
                                                                                      • 4.3.2 Conditional Probability
                                                                                        • 4.3.3 Sample Space Diagrams
                                                                                        • 4.4 Probability Distributions
                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                            • 4.4.2 Expected Values
                                                                                            • 4.5 Binomial Distribution
                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                • 4.6 Normal Distribution
                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                    • 4.7 Hypothesis Testing
                                                                                                      • 4.7.1 Hypothesis Testing
                                                                                                        • 4.7.2 Chi-squared Test for Independence
                                                                                                          • 4.7.3 Goodness of Fit Test
                                                                                                            • 4.7.4 The t-test
                                                                                                          • 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 Integration
                                                                                                                    • 5.2.1 Trapezoid Rule: Numerical Integration
                                                                                                                      • 5.2.2 Introduction to Integration
                                                                                                                        • 5.2.3 Applications of Integration
                                                                                                                      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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