Venn diagrams, sets, vectors and functions. Revision

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

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

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

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?

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

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

Venn diagrams using 3 sets 15 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?

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?

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

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

Vector displacement

Adding vectors

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

Example

Example

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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

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)

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

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)