revision. vectors 1 2 video vid212videovid2 vid3 sets 1 2 vid112vid1 functions 1 vid1 vid21vid1vid2
TRANSCRIPT
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Venn diagrams, sets, vectors and functions.
Revision
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Vectors 1 2 video vid2Vid3Sets 1 2 Vid1Functions 1 Vid1 vid2
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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
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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
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What region has been shaded here?
BA
BA
ξ
Using Correct Notation to Define Regions of a Venn Diagram.
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What region has been shaded here?
BA
BA
ξ
Using Correct Notation to Define Regions of a Venn Diagram.
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What region has been shaded here?
A
BA
ξ
Using Correct Notation to Define Regions of a Venn Diagram.
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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
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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
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What region has been shaded here?
BA
BA
ξ
Using Correct Notation to Define Regions of a Venn Diagram.
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What region has been shaded here?
BA
BA
ξ
Using Correct Notation to Define Regions of a Venn Diagram.
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Vectors - movementThe diagram shows the translation of a triangle by the vector
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Vector displacement
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Adding vectors
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• 6a means 6 lots of vector a
• So if a = then 6a =
What do we mean by 6a?
4
3
24
18
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Example
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Example
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1
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2
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3
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4
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5
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6
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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
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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
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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
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• 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)