Temperature Readings The equation to convert the temperature from degrees Fahrenheit to degrees Celsius is: c(x) = (x - 32) The equation to convert the temperature from degrees Celsius to degrees Fahrenheit is: f(x) = x + 32 Is there a temperature that has the same reading in both Fahrenheit and Celsius? 9 5 5 9
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Temperature Readings The equation to convert the temperature from degrees Fahrenheit to degrees Celsius is: c(x) = (x - 32) The equation to convert the.
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Temperature Readings
The equation to convert the temperature from degrees Fahrenheit to degrees Celsius is:
c(x) = (x - 32) The equation to convert the temperature from
degrees Celsius to degrees Fahrenheit is:
f(x) = x + 32 Is there a temperature that has the same
reading in both Fahrenheit and Celsius? €
9
5€
5
9
FunctionSet of ordered pairs {(x,y)| xX, yY}, 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
Equal Functions: f(x) = g(x)
Identity Function: f(x) = x, idR(7) = 7
Constant Function: f(x) = 3, k(x) = y0
Absolute Value Function: y = |x|
Describing Functions
List of ordered pairs Rule Table Graph Function Diagram Verbal Description
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:
y = 2x
y = -3x + 5
y = x + 32
y = x2
€
9
5
Solve Using Mental Math Strategies
2 18 11 9 12 13 9 15 90 14
3 36 16 14 8 25 2 15 0 12 1 11
Algebra Structures
Set Operation(s) with elements in the set Properties that are true but accepted without
proof (axioms) Definitions Theorems which can be proved using
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
Modular Arithmetic: addition, multiplicationSet Theory Operations: , , –, Matrices: addition, multiplicationFunctions: composition as an operationSymmetries of a Triangle, RectangleComplex Numbers (a + bi): addition,
multiplication
The Game of 50
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) 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)
Matrix Addition
Matrix Multiplication€
a b
c d
⎡
⎣ ⎢
⎤
⎦ ⎥ + =
€
a+ e b + f
c+ g d + h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
a b
c d
⎡
⎣ ⎢
⎤
⎦ ⎥ =
€
a• e+ b • g a • f + b • h
c • e+ d • g c • f + d • h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
e f
g h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
e f
g h
⎡
⎣ ⎢
⎤
⎦ ⎥
€
1,1 1,2
2,1 2,2
⎡
⎣ ⎢
⎤
⎦ ⎥
Matrix Operations
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 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
Examples of Groups
(Z,+) (Q,+) (R,+) (Q+, ) (R+, ) (Zn, +n) for all n ≥ 1
(M2 (Z), +)
More Algebraic Structures 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 +