Foundations

•Basics: Notation, and other things•Algebraic manipulations•Indices, Logs, Roots and Surds•Binomial expansion•Trigonometric functions•Trigonometric identities•Functions•Types of function•Solving equations

Sets of numbers

the set of Natural numbers

,.......3,2,1N

,...2,1,0,1,2... Z the set of Integers

R is the set of Real Numbers

the set of Rational Numbers

NZ,Q qpq

p |

bxRxb

axRxa

bxaRxba

bxaRxba

bxaRxba

bxaRxba

|],(

|),(

|),[

|],(

|,

|),(ba

ba

ba

ba

a

b

Interval notation

Order of OperationsB BracketsO Order: Exponentials, Powers, RootsDMAS}

} Division & Multiplication

Addition & Subtraction

Set notation

is an element ofis not an element ofthe set whose members areIntersectionUnionSubset

FoundationsBasics

Index page

))((22 bababa Difference of two squares

Fraction equivalence

d

cb

ad

cba

ad

acabb

ad

bc

acd

bc

ac

b

a

)(

,Fraction arithmetic

b

ac

b

ca

c

ba

cba

bc

a

cb

ac

b

a

cba

bc

ad

c

d

b

a

d

c

b

ab

acc

b

abd

ac

d

c

b

abce

cdae

be

d

bc

abd

bcad

d

c

b

a

11

1

1Splitting the numerator

c

b

c

a

c

ba

Squaring brackets

22

2

2

))(()(

baba

bababa

Partial fractions

x

B

x

A

xx

bax

))((

FoundationsAlgebraic

manipulation

Index page

Laws of Logarithms

a

xx

xxa

xxn

y

xyx

xyyx

b

ba

e

aa

naa

aaa

aaa

log

loglog

lnlog01log ,1log

loglog

logloglog

logloglog

Laws of Indices

x

xx

yxx

xyyx

yxyx

yxyx

b

a

b

a

aaab

aaa

aa

aaa

aaa

)(

,1 10

Laws of Surds

bababa

b

a

b

abaab

22

,

ya axxy log

cb

cba

cbcb

cba

cb

a

2

)(

))((

)(

Rationalising the denominator

Indices notation

abb ab

a

aaa

a

xxx

xxx

x

1

,1

xy alog

xay

xy

FoundationsIndices, logs, roots & surds

Index page

Sigma Notation

n

r

nfnfffrf0

)()1(....)1()0()(

Binomial coefficients

!)!(

!

rrn

nrn

1!0

!)1()!1(

123).........2)(1(!

nnn

nnnn

Factorial notation

Binomial Theorem

rrnn

r

n barn

ba

0

)(

Binomial coefficients in Pascal’s triangle

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

. . . . .1 n . . . . . . . . . . . .n 1

543210

nnnnnn

nnr ,1 . . . . . . . . . ,1 ,0

General termrrn ba

rn

FoundationsBinomial Expansion

nba )(

Index page

Exact values

313

10tan

021

21

231cos

123

21

210sin

23460

General solutions

)2cos()2cos(cos))12sin(()2sin(sin

)tan(tan

xkxkxxkxkx

xkx

sin

cos

tan

1cot

cos

sintan

sin

1cosec

cos

1sec

Trigonometric ratios

Radians

oc

oc

3.571

180

Complementary angles

,22tancot

seccoseccossin

BAABBABABABA

and

are complementaryangles

S AT C

Symmetries of trig functions

tan)tan(tan)tan(tan)tan(cos)cos(cos)cos(cos)cos(sin)sin(sin)sin(sin)sin(

quadrant4th quadrant 3rdquadrant 2nd

FoundationsTrigonometry

functions

Index page

Addition formulaBABABA sincoscossin)sin(

etc. See formula sheet

Double angle formulae cossin22sin

etc.See formula sheet

Pythagorean identities

1sincos 22 etc.See formula sheet

)(tan 1

22

ab

bar

sinsincoscos)cos(sincos

rrrba

Wave functions - cosine

So

and so

sincosrbra

Squares of sin and cos

)2cos11(2

1sin

)2cos1(2

1cos

2

2

sincoscossin)sin(cossin

rrrba

So

and so

sincosrbra

Wave functions - sine

)(tan 1

22

ab

bar

FoundationsTrigonometry

identities

Index page

Definitions

A function is a rule which associates each member of a set, A, the domain a unique element of a set, B, the range

Composite functions

If then is a composite function. Function is calculated first on and function is calculated on the result.

))(( xgfy y

xg

f

Function diagram

Range Domain

f

Natural DomainThis is the set of all real numbers that can be used in a functionPoints to note:

0,ln ,0, ,0,1

xxxxxx

One-to-one Functions/ inverses

f

A B A B

A one-to-one function has an inverse If then rearrange to write . Interchanging and gives .

1 ff

1 f

)(xfy )(1 yfx

x y )(1 xfy

)(xfy

)(1 xfy xy

Graph of a Function

Domain [a,b]= bx|ax R

Range [d,e]

ey|dy R

e

d

cba

)(

)(

cfe

bfd

FoundationsFunctions

Index page

Modulus Function

0,

0, )(

)(

xx

xxxf

xxf

Linear functionbaxxf )(

b

0a

0a

0a

Quadratic functioncbxaxxf 2)(

0a

0a

nmxacbxaxxf 22 )()(

),( nm

c

acb 42

0

0

0

Polynomial functionsn

nxaxaaxf 10)(0naIf the degree of the

polynomial is i.e.

nxfn ))(deg( ,

where and are polynomials. is a proper fraction if

Otherwise the fraction is improper.

Rational functions

0)(,)(

)()( xh

xh

xgxf

)(xg

)(xf

)(xh

))(deg())(deg( xhxg

Odd & Even Functions

A function is even if

A function is odd if

The graph of an odd function has rotation symmetry about the origin and that of an even function has reflection symmetry in the axis.

)()( xfxf

y

)()( xfxf

FoundationsTypes of Functions

Index page

Quadratic

Factorise or use

a

acbbx

2

42 Polynomial: Remainder

theorem & Factor Theorem

If is a factor of

If then the remainder when is divided by is

)(0)( axaf )(xf

Raf )()(xf )( ax

R

Exponential 1

ab

acdx

acdabxdacbx

da

dacbx

cbx

log

loglog

logloglog loglog)(

loglog )(

)(

Exponential 2

a

nxnan

a

mxmam

nmycbyay

cba

xx

xx

xx

xx

loglogor

loglog

or get toSolve0y soyLet

0222

2

Simultaneous –non linear

Try:1) Solve one equation for (or )

in terms of or for (or ) in terms of and substitute in the other equation

2) Factorise or and set the factors equal to zero.

0),(0),(

yxgyxf

),( yxf ),( yxg

x xy

2x

2y

y

FoundationsSolving

equations

Index page

Foundations

That’s all folks!!!