higher maths 1.2.1 - sets and functions

Higher Maths 1 2 Functions 1

The symbol means ‘is an element of’.

Introduction to Set TheoryIn Mathematics, the word set refers to a group of numbers or other types of elements. Sets are written as follows:

Examples

{ 1, 2, 3, 4,

5, 6 }{ -0.7, -0.2, 0.1

}{ red, green,

blue }

4 { 1, 2, 3, 4, 5 }

7 { 1, 2, 3 }

{ 6, 7, 8 } { 6, 7, 8, 9 }

If A = { 0, 2, 4, 6, 8, … 20 }and B = { 1, 2, 3, 4, 5 } then B A

Sets can also be named using letters:

P = { 2, 3, 5, 7, 11, 13, 17, 19, 23, … }

Higher Maths 1 2 1 Sets and Functions

2

N

W

Z

{ 1, 2, 3, 4, 5,

... }{ 0, 1, 2, 3, 4, 5,

... }{ ... -3, -2, -1, 0, 1, 2,

3, ... }

The Basic Number Sets

QRational numbersIncludes all integers, plus any number which can be written as a fraction.

R √7 π Includes all rational numbers, plus irrational numbers such as or .

Real numbers

C Complex numbersIncludes all numbers, even imaginary ones which do not exist.

Whole numbers

Integers

Natural numbers

N W Z Q R C

23 Q √-1 R

Examples


3

Set Theory and Venn DiagramsVenn Diagrams are illustrations which use overlapping circles to display logical connections between sets.

Blue

Animal

Food

Pig

Blueberry Pie

Blue Whal

e?

Rain

Red

Yellow

Orange Juice

Sun

Strawberries

Aardvark

N

WZ

Q

R

C√82

57


4

Function Domain and RangeAny function can be thought of as having an input and an output.

The ‘input’ is sometimes also known as the domain of the function, with the output referred to as the range.

f (x)domain range

Each number in the domain has a unique output number in the range.

The function

has the domain

{ -2, -1, 0, 1, 2, 3 }Find the range.

Imporant

Example

f (x) = x2 + 3x

f (-2) = 4 – 6 = -2

f (-1) = 1 – 3 = -2

f (0) = 0 + 0 = 0

f (1) = 1 + 3 = 4

f (2) = 4 + 6 = 10

Range = { -2, 0, 4, 10 }


5

Composite FunctionsIt is possible to combine functions by substituting one function into another.

f (x) g (x)

g ( )f (x)

is a composite function and is read ‘ ’.

g ( )f (x)g of f of x

Importa

nt g ( )f (x) f ( )g (x)≠

In

general

Given the

functions

Example

g(x) = x + 3

f (x) = 2 x

an

dfind and

. = 2

( )x + 3

= 2 x + 6

= ( ) + 3

2 x

= 2 x + 3

f ( g (x)) g ( f (x))

g ( f (x))

f ( g (x))


6

= x

Inverse of a FunctionIf a function also works backwards for each output number, it is possible to write the inverse of the function.

f (x)

f (x) = x

2 f (4) = 16

f (-4) = 16

Not all functions have an inverse, e.g.

Every output in the range must have only one input in the domain.

does not have an inverse function.

f (x) = x

2

f (16) = ?

-1

domain range

Note that

f ( )-1 f (x)

f (x)-1

= x

x

andf ( )f (x)-1


7

Find the inverse function for .

Finding Inverse Functions

g (x) = 5 x

3 – 2-1

Example g (x)

3 × 5 – 2

g x

x + 2

√3

÷ 5 + 2

g-1

x

x

x + 25

x + 25

3

-1g (x) =

+ 2

÷ 5

3


8

Graphs of Inverse FunctionsTo sketch the graph of an inverse function

, reflect the graph of the function across

the line .

f (x)-1

f (x) y = x

y = x

f (x)-1f (x)

y = xg (x)-1

g (x)


9

Basic Functions and Graphs

f (x) = axf (x) = a sin bx f (x) = a tan bx

f (x) = ax²f (x) = ax³

f (x) =ax

Linear Functions

Quadratic Functions

Trigonometric Functions

Cubic Functions Inverse Functions


10

Exponential and Logartithmic Functions

f (x) = log xa

1

f (x) = a x

1

(1,a)

(1,a)

is called an exponential function with base .

Exponential Functions

f (x) = a x a

The inverse function of an exponential function is called a logarithmic function and is written as .

f (x) = log xa

Logarithmic Functions


11

Finding Equations of Exponential Functions


12

It is possible to find the equation of

any exponential function by

substituting values of and for

any point on the line.

y = a + bx

2

(3,9)

ExampleThe diagram

shows the

graph of

y = a + b

Find the

values of a

and b.

Substitute (0,2):

x

2 = a + b0

= 1 + bb = 1x y

Substitute (3,9):

9 = a + 13

a = 2

a = 83

y = 2 + 1x

higher maths 1.2.1 - sets and functions

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