Revision. Vectors 1 2 video vid212videovid2 Vid3 Sets 1 2 Vid112Vid1 Functions 1 Vid1 vid21Vid1vid2.

Venn diagrams, sets, vectors and functions.

Revision

Vectors 1 2 video vid2Vid3Sets 1 2 Vid1Functions 1 Vid1 vid2

You should be able to;1. Use language, notation and Venn diagrams to

describe sets and represent relationships between sets

2. Write a vector using correct notation, make calculations with vectors, and find its resultant, modulus and representations in terms of a vector.

3. To learn the vocabulary relating to functions. To learn the different types of functions. To practice defining functions and finding composite functions

Set Notation

• Number of elements in set A n(A)

• “…is an element of …”

• “…is not an element of…”

• Complement of set A A'

• The empty set ∅

• Universal set ξ

• A is a subset of B

• A is a proper subset of B

• A is not a subset of B A B⊄

• Union of A and B A U B

• Intersection of A and B A ∩ B

BA

BA

What region has been shaded here?

BA

BA

ξ

Using Correct Notation to Define Regions of a Venn Diagram.

What region has been shaded here?

BA

BA

ξ

Using Correct Notation to Define Regions of a Venn Diagram.

What region has been shaded here?

A

BA

ξ

Using Correct Notation to Define Regions of a Venn Diagram.

1. How many students are in A but not in B?2. How many students are in sets A and B?3. What is the probability of choosing a student

from set A4. What is the probability of choosing a student

who is not in A or B?5. What is the probability of choosing 2 students

who are in both A and B? 380

1

19

1

20

2

ξ

BA

5

11

2 2

597/20

11/20

Reading a Venn Diagram

1. How many students are in A and C but not in B?2. How many students are ONLY in set C?3. What is the probability of choosing a student from set A4. What is the probability of choosing 2 students who are

both in B?

15 5

ξ

C

A B

18 3

9

7

2

Venn diagrams using 3 sets

What region has been shaded here?

BA

BA

ξ

Using Correct Notation to Define Regions of a Venn Diagram.

What region has been shaded here?

BA

BA

ξ

Using Correct Notation to Define Regions of a Venn Diagram.

Vectors - movementThe diagram shows the translation of a triangle by the vector

Vector displacement

Adding vectors

• 6a means 6 lots of vector a

• So if a = then 6a =

What do we mean by 6a?

4

3

24

18

Example

Example

1

2

3

4

5

6

Substituting numbers into functions

A function can be written as:

Substituting is replacing the x so that,

Check these mentally:

f(3)

f(0)

f( 2)

Try some of these:

a) f(1)=

a) g(3)=

a) h(1)=

b) f(-2)=

b) g(-1)=

b) h(-5)=

2 12

6

3

2

27

4 3 – 3 = 9

4 0 – 3 = –3

4 (– 2) – 3 = –11

34)( xxf

17354)5( f52

12)(

x

xf

23)( 2 xxxg

532)( 2 xxh

Composite functions

A composite function is made up of two or more functions.

fg(x) means take g(x) and put it into f(x).Replace each x in f(x) with the complete g(x).

gf(x) means take f(x) and put it into g(x).Replace each x in g(x) with the complete f(x).

Try some of these:

f(x) 3x 1

g(x) x2 3

h(x) x 2

1.fg(x)

2.gf(x)

3.gh(x)

4.hf(x)

3x² + 8

9x² - 6x + 4

x + 1

√(3x – 3)

𝑓 (𝑥 )=𝑥+3 𝑔 ( 𝑥 )=𝑥2−1

) +3

𝑓𝑔 ( 𝑥 )=𝑥2+2

𝑔𝑓 ( 𝑥 )=(𝑥+3)2−1

𝑔 ( 𝑥 )=𝑥2+6 𝑥+8

Inverse Functions

Another way to do the Inverse Functions is to consider what they do.

The INVERSE function finds the input for a given output.

So if f(x) = 5x – 7

y = 5x – 7

We now need to make x the subject….

So x = (y + 7)/5

The inverse function is written as: f –1(x) = (x + 7)/5

• f(x) = (x – 1)3 g(x) = (x – 1)2 h(x) = 3x + 1 • Work out fg (–1)• Find gh(x) in its simplest form.• Find f-1 (x)