Transcript Chapters 3, 4 and 5

Temperature Readings
The equation to convert the temperature from
degrees Fahrenheit to degrees Celsius is:
5
c(x) = (x - 32)
9
 The equation to convert the temperature from
degrees Celsius to degrees Fahrenheit is:
f(x) = 9 x + 32
5
 Is there a temperature that has the same
reading in both Fahrenheit and Celsius?
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
Function
Set of ordered pairs {(x,y)| xX, yY},
where every element of X is associated
with a unique element of Y.
X is the domain (set of inputs) of the
function.
Y is the range of the function.
The image is the set of outputs.
Some Functions to Remember
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Equal Functions: f(x) = g(x)
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Identity Function: f(x) = x, idR(7) = 7
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Constant Function: f(x) = 3, k(x) = y0
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Absolute Value Function: y = |x|
Describing Functions
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List of ordered pairs
Rule
Table
Graph
Function Diagram
Verbal Description
Examples of Function Rules
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f(x) = -2x + 1
f(x) = x
f(x) = 8
f(x) = x2 + 7
f(x) = 2x
Associate each integer with a number that is
twice the integer.
Composite Functions
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The composition of g with f is the function
g o f = g o f(x) = g(f(x))
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Notice that g o f is obtained by first doing f
and then doing g.
Properties of Some Functions
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One-to-one
A function is one-to-one if it never sends two
elements of the domain to the same element
of the range.
Onto
A function is onto if no element of the range
goes unused.
One-to-One or Onto?
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Temperature functions:
5
c(x) = (x - 32)
9
9
f(x) = x + 32
5
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f(x) = |x|, Domain = R, Range = R
 f(x) = x2, Domain = R, Range = R
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 A function that assigns each word in the English to
the first letter in the word.
 A function that assigns each real number with a
point on the number line.
 y = 2x, Domain = Z, Range = Z
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Inverses in Mathematics
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Inverse Property
Additive Inverse, Multiplicative Inverse (reciprocal)
Inverse Operation
Inverse Function
If a function has an inverse function, then it is 1-1.
If a function is 1-1, then it has an inverse function.
g -1(g(x)) = g (g-1(x)) = x, or g -1o g = g o g -1 = id(x)
Find the inverse function of
each of these functions:
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y = 2x
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y = -3x + 5
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y=
9
x
5
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y = x2
+ 32
Solve Using Mental Math
Strategies
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2  18
11  9
12  13
9  15
90  14
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3  36
16  14
8  25
2  15  0
12  1  11
Algebra Structures
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Set
Operation(s) with elements in the set
Properties that are true but accepted without
proof (axioms)
Definitions
Theorems which can be proved using
axioms, definitions and other theorems
Field Axioms
Associative (+, )
 Identity (+, )
 Inverse (+, )
 Closure (+, )
 Commutative (+, )
 Distributive ( over +)
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Binary Operation(s) on Set S
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A binary operation is a function where every
combination of two elements of set S results
in a unique answer in the set.
M: S  S  S
For example, addition, subtraction and
multiplication with Integers are all binary
operations.
Sets and Operations
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Modular Arithmetic: addition, multiplication
Set Theory Operations: , , –, 
Matrices: addition, multiplication
Functions: composition as an operation
Symmetries of a Triangle, Rectangle
Complex Numbers (a + bi): addition,
multiplication
The Game of 50
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Play with the set of numbers { 1, 2, 3, 4, 5, 6 }.
Player 1 chooses a number from the set.
Player 2 chooses a number from the set and
writes the sum of the two numbers.
The players continue choosing numbers and
writing sums.
The first person to choose a number that
results in a sum of 50 wins the game.
Properties for mod(n)
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Activity 4.22
Activity 4.24
Activity 4.25
Activity 4.26
Which of these properties exist for mod(n),
using the binary operations + and ?
Commutative,
Associative,
Identity, (If so, what is the Identity Element?)
Inverse
Matrices, M2(Z)
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Matrix Addition
a b


c d
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

+
e f 


g h 
a  e b  f 
= 

c  g d  h 
Matrix Multiplication

a

c

b
 
d
e

g

a • e  b • g a • f  b • h 
f 

 = 
c
•
e

d
•
g
c
•
f

d
•
h
h 



1,1

2,1
1, 2 

2, 2
Matrix Operations
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Activity 4.40 - 4.43 (addition)
Activity 4.44 - 4.47, 4.48 (multiplication)
Which of these properties exist for M2(Z),
using the binary operations + and ?
Commutative,
Associative,
Identity, (If so, what is the Identity Element?)
Inverse
Algebraic Structures
Set, Operation(s), Properties
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Group:
A group is a set G together with a binary operation *
which satisfy the following:
(a) The operation * is associative for all elements of G.
(b) G contains a unique identity element, e. If x is any
element of g, e * x = x and x * e = x.
(c) Each element of G has an inverse in G. If x is any
element of g, x-1 is the inverse of x.
x * x-1 = e and x-1 * x = e
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Examples of Groups
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(Z,+)
(Q,+)
(R,+)
(Q+, )
(R+, )
(Zn, +n) for all n ≥ 1
(M2 (Z), +)
More Algebraic Structures
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An Abelian Group is a group (G, *) for which the
operation is commutative.
A Ring is a set R with two operations we will call
addition and multiplication, R(+,).
 A ring has the following properties.
Associative, Commutative, Identity, Inverse for +
(Abelian Group for +)
Associative for 
Distributive of  over +
Examples of Rings: (Z,+, ), (Q,+, ), (R,+, ), (Zn,+, ),
(M2(Z),+, )
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Fields
A field is a set F with two binary operations we will
call addition and multiplication, F(+,).
 A Field has the following properties.
Associative (+, )
Commutative (+, )
Identity (+, )
Inverse (+, x)
(All nonzero elements have an inverse in F.)
Distributive ( over +)
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