Revision

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

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

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

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

What region has been shaded here? 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 A 4.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? ξ BA /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 A 4.What is the probability of choosing 2 students who are both in B? 155 ξ C AB Venn diagrams using 3 sets

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

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

Vectors - movement The 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?

Example

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

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: 3x² + 8 9x² - 6x + 4 x + 1 √(3x – 3)

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 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)