Functions (Including |mod|)

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A2 Maths – Pure – Modulus Function

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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Dr Frost Videos

• Topic Video (Up to 13:44)

Bicen Maths Live Lesson

Functions and Graphs 1 • The Modulus Function • P2 Ex2A • 🚀

Functions and Graphs 2 • Sketching and Solving with Modulus • P2 Ex2A • 🚀

Functions and Graphs 3 • Tricky Modulus Questions • P2 Ex2A Q13 • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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Exam Practice

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Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 570a: Determine the output of a function involving a modulus function.
• 570b: Determine the output of a composite function involving a modulus function.
• 570d: Determine the coordinates of the vertex or intercepts of a modulus graph.
• 570e: Sketch graphs given in the form y = ≤ft|ax + bright|
• 570f: Sketch graphs given in the form y = ≤ft|ax + bright| + c
• 570g: Sketch graphs given in the form y = a – ≤ft|bx + cright|
• 570: Exam Practice: Sketching graphs involving the modulus function

Further Practice

• Practise Questions

Other Videos

Modulus Functions

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Ticktock Maths
  • The Modulus Function lesson (slides)

Worksheets

• Jethwamaths Starter Activities (p1-2)
• kutasoftware Sketching Absolute Graphs
• Solomon C3 Functions Worksheet D✦ (Answers✦)
• PMT PPQs OCR MEI Modulus (Answers)
• Mathshelper (Jon Stone)
  • Modulus Function
• Mathsaurus (Kevin Olding)
  • The modulus function (exam questions)
• Physics & Maths Tutor
  • Modulus of Functions: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• SRWhitehouse
  • Modulus function worksheets✦

A2 Maths – Pure – Mappings

A2 Maths – Pure – Domain and Range

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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2B Part 1 Slides (Courtesy of Owen134866)

Dr Frost Videos

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Bicen Maths Live Lesson

Functions and Graphs 4 • The Definition of a Function • P2 Ex2B • 🚀

Functions and Graphs 5 • Domain and Range • P2 Ex2B • 🚀

Functions and Graphs 6 • Domain and Range summary • P2 Ex2B • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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• Dr Frost Slides 10-13✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 565a: Determine whether a mapping is a function using a mapping diagram. (not yet active)
• 565b: Determine whether a mapping is a function from its graph. (not yet active)
• 565c: Determine whether a function is one-to-one or many-to-one from its graph. (not yet active)
• 565d: Determine whether a basic function is many-to-one or one-to-one from its equation. (not yet active)
• 565e: Determine whether a quadratic with restricted domain is one-to-one vs many-to-one. (not yet active)
• 565f: Determine whether a function with a given domain has an inverse. (not yet active)
• 564a: Determine the domain or range of a sketched function. (not yet active)
• 564b: Determine the value that must be excluded from a function given in the form f(x) = ax+b/cx+d (not yet active)
• 564c: Determine the range of an exponential function f(x)=a+be^kx (not yet active)
• 564d: Determine the range of a trigonometric function for a given domain. (not yet active)
• 564e: Determine the domain of a trigonometric function that would yield a certain range. (not yet active)
• 564f: Determine the allowed domain of a function given in the form √ax + b (not yet active)
• 564g: Determine the range of a quadratic function given in the form x² + k or k – x² (not yet active)
• 564h: Determine the range of a quadratic function by completing the square.
• 564i: Determine the range of a general quadratic function with a restricted domain. (not yet active)
• 564j: Determine the range of a rational function f(x)=ax+b/cx+d
• 564: Exam Practice: Domain and range of more advanced functions (e.g. exponential)
• 565: Exam Practice: One-to-one vs many-to-one functions

Further Practice

• Practise Questions

Other Videos & Multimedia

Functions and Function Notation

Functions and Function Notation

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• MEI – Domain/Range Grid✦, Domain and Range Card Sort✦, What Type of Function?✦
• Region 18 Domain and Range Dominoes✦
• Underground Mathematics Domain and Range Dominoes✦
• colmanweb.com Functions Matching Activity✦
• Math = Love (Sarah Carter)
  • Function or Not Card Sort
  • Discovering the Vertical Line Test
  • Function or Not Puzzle
  • Function Auction Activity
  • Domain & Range Foldable
  • Function or Not Notes
  • Relations, Functions & Dating Advice
  • Vertical Line Test Notes
• Desmos
  • Function or Not Card Sort
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q4
• Ticktock Maths
  • Functions and Mappings lesson (slides)

Worksheets

• Solomon C3 Functions Worksheet A✦ (Answers✦)
• MadAsMaths (T. Madas)
  • Functions Exam Questions Q62, 94, 120, 136, 152 (difficulty 4–5 of 5)
  • Functions Practice Function Introduction Q1–13

A2 Maths – Pure – Solving Equations with Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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2B Part 2 Slides (Courtesy of Owen134866)

Dr Frost Videos

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Bicen Maths Live Lesson

Functions and Graphs 7 • Piecewise Functions • P2 Ex2B • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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• Dr Frost Slides 13-18✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 487a: Use a piecewise function to find an output for a specific input.
• 487b: Sketch a piecewise function involving linear segments.
• 487c: Sketch a piecewise function that includes non-linear segments. (not yet active)
• 487e: Determine the number of solutions to f(x) = k where the function is piecewise.
• 487f: Solve equations given in the form f(x) = k when the function is piecewise.
• 487g: Determine the range of a piecewise function.
• 487h: Determine a piecewise function given knowledge of its range.
• 487: Exam Practice: Piecewise functions

Further Practice

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• kutasoftware
  • Evaluating Functions
  • Piecewise Functions
• MadAsMaths (T. Madas)
  • Functions Exam Questions Q72 (difficulty 4 of 5)

A2 Maths – Pure – Composite Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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2C Slides (Courtesy of Owen134866)

Dr Frost Video

KS5 – Composite Functions

Whole Topic Video

Key Skill – Find the numerical output of a composite function.

K888: Find the numerical output of a composite function.

Bicen Maths Live Lesson

Functions and Graphs 8 • Composite Functions • P2 Ex2C • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 19-23✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 435a: Calculate the numeric output of a composite function.
• 435c: Determine the algebraic output of a composite function where at least one function is quadratic.
• 435d: Determine the algebraic output of a composite function for more general functions. (not yet active)
• 435e: Identify the constituent functions that make up a composite function. (not yet active)
• 435f: Solve an equation involving a composite function, leading to a linear equation. (not yet active)
• 435g: Solve an equation involving a composite function, leading to a quadratic equation. (not yet active)
• 567a: Determine the output of a composite function for a numeric input, where one of the functions contains e^x or ln(x) (not yet active)
• 567b: Find the algebraic output of a composite function (including e^x) (not yet active)
• 567c: Determine the range of a composite function involving e^x or ln(x) (not yet active)
• 567d: Determine values for which fg(x) = gf(x), where one of the functions involves e^x or ln(x) (not yet active)
• 567e: Determine a composite function ff(x) where f(x) is given in the form ax+b/cx+d (not yet active)
• 567f: Determine a coefficient or constant given a known output of a composite function involving e^x or ln(x) (not yet active)
• 567g: Determine the input of a composite function involving adding or subtracting of algebraic fractions. (not yet active)
• 435: Exam Practice: Composite functions (excluding exponentials and modulus)
• 567: Exam Practice: Composite functions involving e^x, ln x or algebraic fractions

Further Practice

• Practise Questions

Other Videos

Composite Functions 2

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Underground Mathematics Compose!✦
• MEI Composite Functions Arithmagons
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q3, Q16, Q17
• Ticktock Maths
  • Composite Functions lesson (slides)

Worksheets

• Jethwamaths Starter Activities (p3-4)
• Solomon C3 Functions Worksheet B✦ (Answers✦)
• Physics & Maths Tutor
  • Composite and Inverse Functions: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• SRWhitehouse
  • Composite and inverse functions worksheets✦
• MadAsMaths (T. Madas)
  • Exponentials & Logarithms Exam Questions Q18, 41 (difficulty 3 of 5)
  • Functions Exam Questions Q6, 14, 18–20, 22–23, 25, 28, 30, 45, 48–49, 52–53, 56–57, 59, 61, 64–65, 70, 73–75, 77, 82, 84, 88, 91, 97–99, 107, 109, 114–115, 119, 125–126, 135, 137 (difficulty 2–4 of 5)
  • Functions Practice Function Compositions Q1–17
  • Trigonometric Inverse Functions Hard Questions Q11 (difficulty 4 of 5)
  • Trigonometry Exam Questions Hard Questions Q76 (difficulty 4 of 5)

A2 Maths Pure – Inverse Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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2D Slides (Courtesy of Owen134866)

Dr Frost Videos

• Topic Video

Bicen Maths Live Lesson

Functions and Graphs 9 • Inverse Functions • P2 Ex2D • 🚀

Functions and Graphs 10 • Inverse Functions with exponentials • P2 Ex2D • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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• Dr Frost Slides 24-30✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions (No Inverse Graphs)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 436b: Determine the inverse function for a linear expression.
• 436c: Determine the inverse function for a expression containing a power or root of x (not yet active)
• 436d: Determine the value of inverse function for a given input. (not yet active)
• 436f: Determine the inverse function where f(x) = ax+b/cx+d
• 436g: Determine the inverse function for a quadratic expression by completing the square.
• 436h: Solve equations of the form f^-1(x) = k (not yet active)
• 436i: Determine the inverse function of a composite function. (not yet active)
• 436j: Solve an equation involving both a composite and inverse function. (not yet active)
• 436k: Simplify f^-1 f(x) and understand self-inverse functions. (not yet active)
• 568a: Determine the inverse of exponential and logarithmic functions.
• 568b: Determine the domain of the inverse function. (not yet active)
• 569a: Sketch the graph of an inverse function given the graph of the original function (no restricted domains and excluding exponential and log graphs). (not yet active)
• 569b: Recognise that the domain of a function may need to be restricted in order to sketch the inverse function. (not yet active)
• 569c: Sketch the graph of an inverse function given the graph of an exponential or logarithmic function. (not yet active)
• 569e: Solve equations given in the form f^-1(a) = f(a) (not yet active)
• 436: Exam Practice: Inverse functions (excluding exponentials and modulus)
• 568: Exam Practice: Inverse functions involving e^x and ln(x)
• 569: Exam Practice: Graphs of inverse functions

Further Practice

• Practise Questions

Other Videos & Multimedia

Inverse Functions 2

Courtesy of Maths Genie

How all Calculators should look

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• eweston Functions Murder Mystery✦ (TES Account Required)
• Andrew Kirby Inverse, Composite, Range and Domain of Functions (TES Account Required)
• Math = Love (Sarah Carter)
  • Equations of Inverses Foldable
  • Sketching Inverses Practice Book
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q5
• Ticktock Maths
  • Inverse Functions lesson (slides)

Worksheets

• kutasoftware Function Inverses
• Jethwamaths Starter Activities (p5-6)
• Solomon C3 Functions Worksheet C✦ (Answers✦)
• PMT PPQs OCR MEI Composite & Inverse Set 1 (Answers), Set 2 (Answers), Set 3 (Answers)
• martahidegkuti
  • Inverse functions – notes and problems✦
• MadAsMaths (T. Madas)
  • Exponentials & Logarithms Exam Questions Q11, 35, 39, 74 (difficulty 2–4 of 5)
  • Functions Exam Questions Q2–5, 9, 11, 13, 16–17, 27, 33–36, 38–41, 43–44, 46–47, 51, 63, 66, 68, 71, 78, 81, 83, 85–87, 89–90, 95–96, 108, 112, 116–117, 123, 127, 133, 151 (difficulty 2–5 of 5)
  • Functions Practice Function Inverses Q1–23

Other Summary Sites

• onlinemathlearning

A2 Maths – Pure – Modulus Graphs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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2E Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Functions and Graphs 11 • Further Transformations • P2 Ex2E/F • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths (Up to 15:30)

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 31-34✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 570c: Sketch a graph of y = ≤ft| f(x) right| given a sketch of y = f(x)
• 570h: Sketch a modulus graph of a quadratic function.
• 570i: Sketch a modulus graph of a trigonometric function.
• 570j: Determine the range of a modulus function.
• 570k: Use a sketch of a modulus function to determine its equation.
• 570l: Sketch a graph of y = f(≤ft|xright|) given a sketch of y = f(x)
• 570: Exam Practice: Sketching graphs involving the modulus function

Other Videos

Transforming Graphs of Modulus Functions

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Modulus Triples – Teacher Notes✦ and Activity✦ – Kangaroo Maths
• eweston Modulus Always/Sometimes/Never✦ (TES Account Required)
• MeadowsJ Modulus Graphs Picture Challenges✦ (with Autograph or Desmos) (TES Account Required)
• Math = Love (Sarah Carter)
  • Absolute Value Graphs Foldable
  • Describing Absolute Value Graphs
  • Graphing Absolute Value Foldable
  • Absolute Value Functions Notes
• Ticktock Maths
  • Modulus and Graphs lesson (slides)

A2 Maths – Pure – Multiple Graph Transformations

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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2F Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Functions and Graphs 11 • Further Transformations • P2 Ex2E/F • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths (From 15:30)

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 35-39✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 451a: Sketch the graph y = f(x + a) + b given the graph of y = f(x)
• 451b: Sketch the graph y = af(bx) given the graph of y = f(x) (not yet active)
• 451c: Sketch the graph af(x)+b, af(x+b) and f(ax)+b given the graph of y = f(x)
• 451h: Determine the equation in f(x) notation after a translation of a given graph to y = f(x + a) + b (not yet active)
• 451i: Determine the equation in f(x) notation after the transformation of a given graph to y = af(bx) (not yet active)
• 451j: Determine the equation in f(x) notation after the transformation of a given graph to y = af(x)+b, y = af(x+b) or y = f(ax)+b (not yet active)
• 451l: Determine the new equation of a function after a translation by a given vector beginpmatrixabendpmatrix (not yet active)
• 451m: Determine the new equation of a function after transformations given in the form y = af(bx) (not yet active)
• 451n: Determine the new equation of a function after a combination of transformations, y = af(x)+b, y = af(x+b) and y = f(ax)+b (not yet active)
• 451: Exam Practice: Combinations of graph transformations

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions 1 · 2
• The Chalk Face Graph Transformation Challenge
• Dan Meyer Desmos Activities (1, 2, 3 & 4)
• Math = Love (Sarah Carter)
  • Absolute Value Transformations Foldable
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q2
• Jo Morgan
  • Transformations of y = lnx✦
• Ticktock Maths
  • Combining Transformations lesson (slides)

Worksheets

• Jethwamaths Starter Activities (p7-8)
• kutasoftware Transformations of Graphs
• Solomon C3 Functions Worksheet E✦ (Answers✦), Worksheet F✦ (Answers✦)
• PMT PPQs OCR MEI Combined Transformations (Answers)
• MadAsMaths (T. Madas)
  • Reciprocal Functions Q9–11, 21 (difficulty 3–4 of 5)
  • Curve Sketching Exam Questions Q14 (difficulty 5 of 5)
  • Exponentials & Logarithms Exam Questions Q85, 91, 101 (difficulty 4 of 5)
  • Functions Exam Questions Q113, 131–132 (difficulty 4 of 5)
  • Modulus Function Exam Questions Q37, 64, 67–69, 76, 83, 95 (difficulty 3–5 of 5)
  • Transformations Of Graphs Exam Questions Q13, 20, 30, 38, 44–45, 49, 53–54, 61, 64, 67, 69, 71–72, 74, 76, 79, 82, 91 (difficulty 2–5 of 5)
  • Transformations Of Graphs Practice Q2–3, 5–6 · questions only
  • Curve Sketching Q35, 59, 66, 75, 101 (difficulty 4–5 of 5)
• Starting Points Maths
  • Transformations of Functions: 2 Steps · PDF

See Also:

• WW AS Sketching & Transforming Graphs

A2 Maths – Pure – Solving Modulus Problems

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

2G Slides (Courtesy of Owen134866)

Dr Frost Videos

• Topic Video

Bicen Maths Live Lesson

Functions and Graphs 12 • Solving Modulus Problems • P2 Ex2G • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 40-44✦

Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 571a: Solve an equation given in the form ≤ft|ax + bright| = c
• 571b: Determine the roots of a modulus graph.
• 571c: Solve an equation given in the form ≤ft|ax+bright|=cx+d
• 571d: Solve an equation given in the form ≤ft|ax+bright| = k≤ft|cx + dright|
• 571e: Determine a coefficient or constant in a modulus equation given the number of intersections with another graph.
• 571f: Solve a modulus equation involving an exponential function.
• 571g: Solve a modulus equation involving a quadratic.
• 571h: Determine the inverse of modulus functions. (not yet active)
• 572a: Solve a modulus inequality where the graphs are already sketched.
• 572b: Solve modulus inequalities given in the form ≤ft|ax + bright| > c
• 572c: Solve modulus inequalities given in the form ≤ft|ax + bright| > cx + d
• 572d: Solve modulus inequalities given in the form ≤ft|ax + bright| > ≤ft|cx + dright|
• 572e: Solve modulus inequalities involving algebraic constants.
• 572f: Solve modulus inequalities involving a quadratic.
• 571: Exam Practice: Solving equations involving the modulus equation
• 572: Exam Practice: Solving inequalities involving the modulus function

Further Practice

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Andy Lutwyche
  • Inequalities Codebreaker
• Math = Love (Sarah Carter)
  • Leveled Practice Cards
  • Absolute Value Equations Organisers
  • Guest Post: Solving Absolute Value
  • Absolute Value Equations Foldable
• Ticktock Maths
  • Solving Modulus Problems lesson (slides)

External Whole Topic Past Paper Style Questions

• WW Dr Frost Style QQQs (Version 1) Word Doc, PDF (Q’s Only) or PDF (Q’s & Sol.)
• WW Dr Frost Style QQQs (Version 2) Word Doc, PDF (Q’s Only) or PDF (Q’s & Sol.)
• Jethwamaths Worksheet, Topic Test & Starter Activities
• Mathsaurus Mixed Exam Board PPQs Functions, Modulus Function
• Jo Hill Desmos Exam Questions
• Naikermaths Functions PPQs✦ (Solutions✦), Graph Transformations✦ (Solutions✦) 
• Solomon C3 Functions Worksheet G✦ (Answers✦), Worksheet H✦ (Answers✦)
• PMT PPQs by Topic – note most of these extend into Binomial Expansion or Integration
  • AQA
  • Edexcel
    • Modulus Functions Set 1✦, Set 2
    • Transformations and Graphs Set 1✦, Set 2
  • OCR Functions✦
  • OCR MEI
    • The Language of Functions
    • The Modulus Function
  • Cheat Sheet
• Dr Frost Topic Test✦ (Teacher Login Required)
• Transum
  • Exam-style questions: Functions
• Susan Wall (Active A-level Maths)
  • Properties of Functions: Basic function notation · Understanding function notation · Exploring functions (Susan Wall;

Other Whole Topic Resources

• Berwick Maths Diagnostic Questions

Activities

• NRICH (student problems)
  • Algebra & Functions
• MEI
  • MEI teaching notes, misconceptions & prompts: Functions✦
• MathsPad
  • Four functions✦
• Andy Lutwyche
  • Erica’s Errors Graphs

Worksheets

• MathedUp
  • Functions: Questions · Challenge · Answers
• Maths Genie
  • Functions and Graphs: Questions · Solutions · Lesson · Topic page
• Mathsaurus (Kevin Olding)
  • Functions, inverses, domain and range (exam questions)
• MME
  • C3 Functions: Questions | Answers
• SRWhitehouse
  • Functions worksheets✦
• begbiesan
  • Functions tests✦
• eweston
  • Maths Vegas functions review✦
• MadAsMaths (T. Madas)
  • Functions Exam Questions Q1, 7–8, 10, 12, 15, 21, 24, 29, 31–32, 37, 42, 50, 55, 60, 76, 79–80, 92–93, 100, 104, 124, 128 (difficulty 2–4 of 5)
  • Modulus Function Exam Questions Q49, 57, 60, 97 (difficulty 3–5 of 5)
  • Transformations Of Graphs Exam Questions Q18 (difficulty 2 of 5)