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Cambridge IGCSE and O Level Additional Mathematics
Cambridge IGCSE and O Level Additional Mathematics
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,ContentsIntroduction iv
CHAPTER 1 Functions 1
CHAPTER 2 Quadratic functions 19
CHAPTER 3 Equations, inequalities and graphs 41
CHAPTER 4 Indices and surds 58
CHAPTER 5 Factors of polynomials 68
CHAPTER 6 Simultaneous equations 80
CHAPTER 7 Logarithmic and exponential functions 90
CHAPTER 8 Straight line graphs 114
CHAPTER 9 Circular measure 128
CHAPTER 10 Trigonometry 138
CHAPTER 11 Permutations and combinations 173
CHAPTER 12 Series 184
CHAPTER 13 Vectors in two dimensions 210
CHAPTER 14 Differentiation 225
CHAPTER 15 Integration 257
CHAPTER 16 Kinematics 277
Mathematical notation 290
Answers 293
Index 334
,Functions
If A equals success, then the formula is A equals X plus Y plus Z, with X
being work, Y play, and Z keeping your mouth shut.
Albert Einstein (1879 – 1955)
point
You can think of the display as a mapping. Some of the values are shown below.
Amount of fuel (litres) Cost ($)
1 2.40
2 4.80
3 7.20
4 9.60
5 12.00
10 24.00
50 120.00
100 240.00
If x is an element of the first set, then f(x) denotes the associated
element from the second set. For example, this mapping diagram shows
integers mapped onto the final digit of their squares.
Input Output
1 1
x
1
, 1 FUNCTIONS
point
A function is a rule that associates each element of one set (the input)
with only one element of a second set (the output). It is possible for
more than one input to have the same output, as shown above.
You can use a flow chart (or number machine) to express a function.
This flow chart shows a function, f, with two operations. The first
operation is and the second operation is 3.
Input Output
You can write the equation of a line in the form y 2x 3 using
Read this as ‘f of function notation.
x equals two x plus f(x) 2x 3 Read this as ‘f maps x
three.’ or f: x 2x 3 onto two x plus three.’
Using this notation, you can write, for example:
f(4) 2 4 3 11
or f:(5) 2 (5) 3 7
The domain and range
The domain of a function f(x) is the set of all possible inputs. This is the set of
Real numbers are values of x that the function operates on. In the first mapping diagram of the
all of the rational next worked example, the domain is the first five positive odd numbers. If no
and irrational domain is given, it is assumed to be all real values of x. This is often denoted
numbers. by the letter ℝ.
The range of the function f(x) is all the possible output values, i.e. the
corresponding values of f(x). It is sometimes called the image set and is
controlled by the domain.
In certain functions one or more values must be excluded from the
domain, as shown in the following example.
Worked example
1
For the function f(x) 2 x 1:
a Draw a mapping diagram showing the outputs for the set of inputs odd
numbers from 1 to 9 inclusive.
b Draw a mapping diagram showing the outputs for the set of inputs even
numbers from 2 to 10 inclusive.
c Which number cannot be an input for this function?
2
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