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

Examples of Function Rules

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

The composition of g with f is the function

g o f = g o f(x) = g(f(x))

Notice that g o f is obtained by first doing f and then doing g.

Properties of Some Functions

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?

Temperature functions:c(x) = (x - 32)

f(x) = x + 32

f(x) = |x|, Domain = R, Range = R f(x) = x2, Domain = R, Range = R 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

€

9

5€

5

9

Inverses in Mathematics

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:

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

axioms, definitions and other theorems

Field Axioms

Associative (+, )Identity (+, )Inverse (+, )Closure (+, )Commutative (+, )Distributive ( over +)

Binary Operation(s) on Set S

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 +

Examples of Rings: (Z,+, ), (Q,+, ), (R,+, ), (Zn,+, ), (M2(Z),+, )

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 +)