1. Venn diagrams, sets, vectors and functions.
Revision

2. You should be able to;
Use language, notation and Venn diagrams to describe sets and represent relationships between sets
Write a vector using correct notation, make calculations with vectors, and find its resultant, modulus and representations in terms of a vector.
To learn the vocabulary relating to functions. To learn the different types of functions. To practice defining functions and finding composite functions

3. 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

4. Using Correct Notation to Define Regions of a Venn Diagram.B ξ
What region has been shaded here?

5. What region has been shaded here?
Using Correct Notation to Define Regions of a Venn Diagram. A B ξ
What region has been shaded here?

6. Using Correct Notation to Define Regions of a Venn Diagram.B ξ
What region has been shaded here?

7. Reading a Venn Diagram How many students are in A but not in B?ξ B A 5 2 2 11
How many students are in A but not in B?
How many students are in sets A and B?
What is the probability of choosing a student from set A
What is the probability of choosing a student who is not in A or B?
What is the probability of choosing 2 students who are in both A and B? 5 9
7/20
11/20

8. Venn diagrams using 3 sets15 5 ξ C A B 1 8 3 9 7 2
How many students are in A and C but not in B?
How many students are ONLY in set C?
What is the probability of choosing a student from set A
What is the probability of choosing 2 students who are both in B?

9. What region has been shaded here?
Using Correct Notation to Define Regions of a Venn Diagram. A B ξ
What region has been shaded here?

10. Using Correct Notation to Define Regions of a Venn Diagram.B ξ
What region has been shaded here?

15. Vectors - movement
The diagram shows the translation of a triangle by the vector

16. Vector displacement

17. Adding vectors

18. What do we mean by 6a?
6a means 6 lots of vector a
So if a = then 6a =

19. Example

20. Example

21. 1

22. 2

23. 3

24. 4

25. 5

26. 6

29. Substituting numbers into functions
A function can be written as:
Try some of these:
Substituting is replacing the x so that,
a) f(1)= 2
b) f(-2)= 12
Check these mentally:
4  3 – 3 = 9
a) g(3)= 2
b) g(-1)= 6
4  0 – 3 = –3
4 (– 2) – 3 = –11
a) h(1)= 27
b) h(-5)= 3

30. Composite functions
A composite function is made up of two or more functions.
Try some of these:
𝑓 𝑥 =𝑥+3
𝑔 𝑥 = 𝑥 2 −1
fg(x) means take g(x) and put it into f(x).
Replace each x in f(x) with the complete g(x).
3x² + 8
𝑓𝑔 𝑥 = (𝑥 2 −1) +3
9x² - 6x + 4
𝑓𝑔 𝑥 = 𝑥 2 +2
gf(x) means take f(x) and put it into g(x).
Replace each x in g(x) with the complete f(x).
x + 1
𝑔𝑓 𝑥 = (𝑥+3) 2 −1
𝑔 𝑥 = 𝑥 2 +6𝑥+8
√(3x – 3)

31. Inverse Functions So if f(x) = 5x – 7 y = 5x – 7
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

33. 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)