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CIE IGCSE Maths: Extended
Revision Notes
Home / IGCSE / Maths: Extended / CIE / Revision Notes / 2. Algebra & Graphs / 2.10 Functions / 2.10.1 Functions Toolkit
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2.10.1 Functions Toolkit
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1. NUMBER
Introduction to Functions
What is a function?
2. ALGEBRA & GRAPHS
A function is a combination of one or more mathematical operations that takes a
set of numbers and changes them into another set of numbers
It may be thought of as a mathematical “machine”
Putting 3 in to the function would give 2 × 3 + 1 = 7
Putting -4 in would give 2 × (-4) + 1 = -7
Putting x in would give 2x ! 1 For example, if the function (rule) is “double
2.1 Algebra Toolkit
2.2 Algebraic Roots & Indices
the number and add 1”, the two mathematical operations are "multiply by 2
(×2)" and "add 1 (+1)"
The number being put into the function is often called the input
The number coming out of the function is often called the output
2.3 Expanding & Factorising
Brackets
What does a function look like?
2.4 Linear Equations &
Inequalities
A function f can be written as f(x) = … or f : x ↦ …
These two different types of notation mean exactly the same thing
Other letters can be used. g, h and j are common but any letter can
technically be used
Normally, a new letter will be used to define a new function in a
question
For example, the function with the rule “triple the number and subtract 4” would
be written
f !x" ! 3x – 4 or f:x ! 3x – 4
2.5 Quadratic Equations
2.6 Rearranging Formula
2.7 Simultaneous Equations
In such cases, x would be the input and f ! x " would be the output
Sometimes functions don’t have names like f and are just written as y = …
eg. y ! 3x – 4
2.8 Algebraic Fractions
2.9 Forming & Solving
Equations
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2.10 Functions
2.10.1 Functions Toolkit
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%#&5(&'(*%"($&(('"%%"%$#,$3&6"$7+"76,8#%$"&%*"'
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2.10.2 Composite & Inverse
Functions
!""#$%&'()*&+$(,-"./
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2.11 Sequences
How does a function work?
2.12 Coordinate Geometry
! x " and output !f ! x " or y"
Whatever goes in the bracket (instead of x )with f, replaces the x on the other side
A function has an input
This is the input
If the input is known, the output can be calculated
For example, given the function f !x" ! 2x "
2.13 Linear Graphs
1
f !3" ! 2 " 3 # 1 ! 7
f ! ! 4" " 2 # ! ! 4" $ 1 " ! 7
f !a" ! 2a " 1
2.14 Quadratic Graphs
2.15 Further Graphs &
Tangents
If the output is known, an equation can be formed and solved to find the input
For example, given the function f !x" ! 2x " 1
If
f !x" ! 15, the equation 2x " 1 ! 15 can be formed
2.16 Solving & Graphing
Inequalities
Solving this equation gives an input of 7
2.17 Real-Life Graphs
? Worked Example
A function is defined as
(a)
f ! x " ! 3x 2 " 2x # 1 .
2.18 Differentiation
Find f !7" .
The input is x !
3. GEOMETRY
7 , so substitute 7 into the expression everywhere you
see an x .
4. PROBABILITY &
STATISTICS
f !7" ! 3!7" 2 " 2!7" # 1
Calculate.
f !7" ! 3! 49" " 14 # 1
! 147 " 14 # 1
f ! 7"" ! 134
(b)
Find f ! x !
3" .
The input is x !
x " 3 so substitute x ! 3 into the expression
everywhere you see an x .
f ! x # 3" ! 3! x # 3" 2 " 2! x # 3" # 1
Expand the brackets and simplify.
f ! x # 3" ! 3! x 2 # 6x # 9" " 2! x # 3" # 1
! 3x 2 # 18x # 27 " 2x " 6 # 1
! 3x 2 # 16x # 22
f ! x " 3"" ! 3x 2 " 16x " 22
A second function is defined g :
(c)
x ! 3x – 4 .
Find the value of x for which g :
x ! " 16 .
Form an equation by setting the function equal to -16.
3x ! 4 " ! 16
Solve the equation by first adding 4 to both sides, then dividing by 3.
3x ! 4 " ! 16
3x " ! 12
12
x " !
3
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