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"Full Coverage": Functions
This worksheet is designed to cover one question of each type seen in past papers, for each
GCSE Higher Tier topic. This worksheet was automatically generated by the DrFrostMaths
Homework Platform: students can practice this set of questions interactively by going to
www.drfrostmaths.com/homework, logging on, Practise β Past Papers/Worksheets (or Library
β Past/Past Papers for teachers), and using the βRevisionβ tab.
Question 1 Categorisation: Determine the output for a function given an input.
[Edexcel GCSE(9-1) Mock Set 1 Autumn 2016 3H Q20a]
For all values of π₯
π(π₯) = 2π₯ β 3 and π(π₯) = π₯2 + 2
(a) Find π(β4)
π(β4) = ..........................
Question 2 Categorisation: Determine the input that would result in an output.
Given that
π(π₯) = 6π₯ + 1
Find the value of π₯ when
π(π₯) = 7
π₯ = ..........................
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Question 3 Categorisation: Determine the input that would result in a given output, where the
function involves algebraic fractions.
[Edexcel IGCSE May2015-3H Q17c]
π(π₯) =3
π₯ + 1+
1
π₯ β 2
Find the value of π₯ for which π(π₯) = 0 Show clear algebraic working.
π₯ = ..........................
Question 4 Categorisation: Determine the input that would result in a given output, requiring
use of the quadratic formula.
[Edexcel GCSE(9-1) Mock Set 2 Spring 2017 3H Q21b]
π(π₯) =1
π₯ + 2+
1
π₯ β 3
Given that π(π₯) = 4
(c) find the possible values of π₯ .
Give your answer in the form πΒ±βπ
π where π , π and π are positive integers.
..........................
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Question 5 Categorisation: Find the output expression for a function when some arbitrary input
expression is used, e.g. if π(π) = ππ + π, then π(ππ) = πππ + π and
π(π + π) = π(π + π) + π.
[Edexcel Specimen Papers Set 1, Paper 2H Q18]
π(π₯) = 3π₯2 β 2π₯ β 8
Express π(π₯ + 2) in the form ππ₯2 + ππ₯
π(π₯ + 2) = ..........................
Question 6 Categorisation: As Question 5, but used to form a quadratic equation.
Let π(π₯) = π₯2 + π₯
Solve π(π₯ β 1) + π(π₯ + 1) = 6
..........................
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Question 7 Categorisation: Use a graph representation of a function to find an output for a
specified input.
[Edexcel IGCSE Jan2016-3H Q15a]
The diagram shows the graph of π¦ = π(π₯) for β3.5 β€ π₯ β€ 1.5
Find π(0)
π(0) = ..........................
Question 8 Categorisation: Using a graph representation of a function, reason about the
input(s) that result in a specified output.
[Edexcel IGCSE Jan2016-3H Q15b]
The diagram shows the graph of π¦ = π(π₯) for β3.5 β€ π₯ β€ 1.5
For which values of π does the equation π(π₯) = π have only one solution?
..........................
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Question 9 Categorisation: Find the output of a composite function for a numerical input.
[Edexcel IGCSE May2016-4H Q17b]
π is the function such that π(π₯) = 2π₯ β 5
π is the function such that π(π₯) = π₯2 β 10
Find ππ(β4)
..........................
Question 10 Categorisation: Apply the same function multiple times, e.g. ππ(π) or ππ(π).
[Edexcel IGCSE May2016(R)-3H Q18b]
π is the function such that
π(π₯) =π₯
3π₯ + 1
Find ππ(β1)
..........................
Question 11 Categorisation: Determine the input that would result in the output of a composite
function.
[Edexcel IGCSE May2016-4H Q17d]
π is the function such that π(π₯) = 2π₯ β 5
π is the function such that π(π₯) = π₯2 β 10
Solve ππ(π₯) = β1
..........................
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Question 12 Categorisation: Use a graph representation of a function combined with another
algebraically specified function to determine the output of a composite function for
a given input.
[Edexcel IGCSE Jan2016-3H Q15e]
The diagram shows the graph of π¦ = π(π₯) for β3.5 β€ π₯ β€ 1.5
π(π₯) =1
2 + π₯
Find ππ(β3)
ππ(β3) = ..........................
Question 13 Categorisation: Determine a composite function in terms of π.
[Edexcel GCSE(9-1) Mock Set 1 Autumn 2016 3H Q20b Edited]
For all values of π₯
π(π₯) = 2π₯ β 3 and π(π₯) = π₯2 + 2
(b) Determine ππ(π₯) , giving your expression in its simplest form.
ππ(π₯) = ..........................
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Question 14 Categorisation: Appreciate in general that ππ(π) is not necessarily ππ(π), and
potentially use a combination of both to solve equations.
[Edexcel GCSE(9-1) Mock Set 1 Autumn 2016 3H Q20c]
For all values of π₯
π(π₯) = 2π₯ β 3 and π(π₯) = π₯2 + 2
(c) Solve ππ(π₯) = ππ(π₯)
..........................
Question 15 Categorisation: Determine a composite function in terms of π, involving multiple
applications of the same function.
[Edexcel Specimen Papers Set 2, Paper 2H Q9c]
The functions π and π are such that
π(π₯) = 3(π₯ β 4) and π(π₯) =π₯
5+ 1
Find ππ(π₯) , simplifying your expression.
ππ(π₯) = ..........................
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Question 16 Categorisation: Deal with constants in functions,
e.g. if π(π) = ππ + π then π(π) = ππ + π
[Edexcel New SAMs Paper 3H Q10b]
The function π is such that
π(π₯) = 4π₯ β 1
The function π is such that
π(π₯) = ππ₯2 where π is a constant.
Given that ππ(2) = 12 , work out the value of π
π = ..........................
Question 17 Categorisation: Determine the inverse of simple linear functions.
[Edexcel IGCSE May2016-4H Q17c]
π is the function such that π(π₯) = 2π₯ β 5
π is the function such that π(π₯) = π₯2 β 10
Express the inverse function πβ1 in the form πβ1(π₯) = ...
πβ1(π₯) = ..........................
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Question 18 Categorisation: Determine the inverse of functions involving fractions.
[Edexcel Specimen Papers Set 2, Paper 2H Q9b]
The functions π and π are such that
π(π₯) = 3(π₯ β 4) and π(π₯) =π₯
5+ 1
Find πβ1(π₯)
πβ1(π₯) = ..........................
Question 19 Categorisation: Determine the inverse of a function where the subject appears
multiple times.
[Edexcel IGCSE Jan2016(R)-3H Q16c]
π(π₯) =2π₯
π₯ β 1
Find πβ1(π₯)
πβ1(π₯) = ..........................
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Question 20 Categorisation: Appreciate the functions are only defined for certain inputs, e.g.
π(π) =π
π is not defined for 0, π(π) = βπ is not defined for negative inputs. IGCSE
requires understanding of the terms βdomainβ and βrangeβ.
[Edexcel GCSE(9-1) Mock Set 2 Spring 2017 3H Q21b]
π(π₯) =1
π₯ + 2+
1
π₯ β 3
(b) Write down a value of π₯ for which π(π₯) is not defined.
..........................
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Answers
Question 1
π(β4) = 18
Question 2
π₯ = 1
Question 3
π₯ =5
4
Question 4
3Β±β101
4
Question 5
π(π₯ + 2) = 3π₯2 + 10π₯
Question 6
π₯ = β2 or π₯ = 1
Question 7
π(0) = 5
Question 8
π < 5 or π > 13
Question 9
7
Question 10
1
5
Question 11
π₯ = 1 or π₯ = 4
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Question 12
ππ(β3) = 9
Question 13
ππ(π₯) = 4π₯2 β 12π₯ + 11
Question 14
π₯ = 1 or π₯ = 5
Question 15
ππ(π₯) = 9π₯ β 48
Question 16
π =13
16
Question 17
πβ1(π₯) =1
2(π₯ + 5)
Question 18
πβ1(π₯) = 5(π₯ β 1)
Question 19
πβ1(π₯) =π₯
π₯β2
Question 20
"-2 OR 3"