Algebra 2 Hn 1

ALGEBRA II

HELPFUL NOTES 1.1 Number Sets

Natural numbers 0 Whole numbers 0 Integers 0 Rational numbers 0 Real numbers 0 Irrational numbers 0 SET RELATIONSHIPS:

Name all the sets the given number belongs to: 7 16 − 36 12 − 12 36

ALGEBRA II

HELPFUL NOTES 1.2

Properties

Commutative Associative

ADDITION a+b=b+a

MULTIPLICATION a*b=b*a

 a  b  c  a   b  c

 a b  c  a  b c 

Identity

a0 a

1 a  a

Inverse

a  (a )  0

1 a 1 a

Distributive property of addition over multiplication a(b  c)  ab  ac Distributive property of subtraction over multiplication a(b  c)  ab  ac

Fill in the blanks with the property examples below: 3(9  4)  3  9  3  4 2  4  4  (2) 1 0  (8)  8   (5)  1 5 2 5 10   8  (2  4)  6  (8  2)  (4  6) 3 5 15 Addition: Multiplication: Commutative: _____________________ commutative: _____________________ (7  3) 11  7  (3 11) 34  ( 34)  0 37  29  29  37

Associative:

_____________________

associative:

_____________________

Identity:

_____________________

identity:

_____________________

Inverse:

_____________________

inverse:

______________________

Distributive of multiplication over addition or subtraction: _______________________

ALGEBRA II

HELPFUL NOTES 1.3 Axioms

Reflexive

Equations A=A

Symmetric

If A = B, than A = B

Comparison Transitive Addition Multiplication

If A=B and B=C then A=C If A=B, then A+C=B+C If A=B, then A  C=B  C

Inequalities

The only one can be true:A>B, A0 then A  C0 then A  C
1. Name the property: a) Either Alma is younger than Iris, or they are the same age, or Alma is older than Iris.

______________

b) If Ethan is the same age as Kyle, then Kyle is the same age as Ethan.

______________

c) Cody and his dad work for the same company, which gives it’s employees Christmas bonus 10% of the monthly salary. If dad’s salary is higher than Cody’s, then dad’s bonus will be higher than Cody’s.

______________

d) If y + 5 = x – 2 and x – 2 = 10 then y + 5 = 10

______________

e) If y + 5 = 10 , then y + 5 + -5 = 10 + -5, so y = 5

______________

f) If

x x  (2) > 10  (2) , so x > -5 < 10, then 2 2

2. Fill in appropriate sign into the blanks: a) b) c) d)

if if if if

m = n and p < 0, than m = n and p > 0, than m < n and p < 0, than m < n and p > 0, than

ALGEBRA II

mp ____ np, mp ____ np, mp ____ np, mp ____ np,

and m + p ____ n + p and m + p ____ n + p and m + p ____ n + p and m + p ____ n + p

HELPFUL NOTES 1.4 Relations & Functions

______________

Relation: given 4 ways: a) __________

b) __________

c) __________

d) __________

5

5

Domain: input

Range: output

Function: for every input, there is exactly one output Table: It’s a function if x -2 -1 0 1 y 4 1 0 1

________________________ . (-4, 4) (-3, 4) (-2, 4) x 4 1 0 1 (-1, 4) (0, 4) (1, 4) (2, 4) y -2 -1 0 1

Graphs: no vertical line goes through 2 points

Bubbles: write each arrow as a coordinate, then use the rule for coordinates Evaluating: *evaluate f(-2) **evaluate f(x) for x = -2

0 2

3 4

Zeros Zero of an equation is an x value of a point with y = 0 , which is just the function’s ____________________ , the point where the line ______________ x-axis Example 1: GRAPH

Example 2: TABLE X Y -1 15 0 10 2 5 3 0 5 -5 6 -10

Example 3: FUNCTION Slope-intercept form: F(x) = 3x – 9 Not a slope-intercept form: 2x + 7y = 14

Domain and range of functions given by equations The domain of a function (what X can be) is all real numbers – unless the x part is either.… •

In the denominator of some fraction. Ex1: Domain of y =

a is x ≠ 0 x

Under a radical. Ex2: Domain of y = x is x ≥ 0 • A logarithmic function •

The Range of a function is all real numbers - unless the function is a • polynomial with the highest power ________________. • rational, exponential, or absolute value function Find:

2x + 3 a) domain of f(x) = x−2

d) range of f(x) = 2 x 2 + 5

b) domain of f(x) =

e) domain of f(x) = 2 x 2 + 5

2x − 6

c) domain of f(x) = 2x – 6

f) range of f(x) = 2 x 3 + 5

ALGEBRA II

HELPFUL NOTES 1.5 Operations with functions

Given functions

f ( x) = x 2 + x and

ADDITION

f(x) + g(x)

SUBTRACTION

f(x) – g(x)

MULTIPLICATION

f(x) ∗ g(x) f ( x) g ( x)

DIVISION

Given functions f ( x) = 2 x 2 − 3x + 4 and 1) find 3f(x) + 2g(x)

2) find

g ( x) = 2 x find:

g 2 ( x)

3) evaluate f(x) ∗ g(x)

4) evaluate

f ( x) g ( x)

g ( x) = 7 x − 5

ALGEBRA II

HELPFUL NOTES 1.6 Composite functions f(g(x)) = f  g(x)

SYMBOLS:

g(f(x)) = g  f(x) 1. Given

f(x) = 3x 2 and g(x) = 2x + 1

a) evaluate f(g(4))

c) find f  g(x)

b) evaluate g(f(4))

d) find g  f(x)

2. Given

1 f ( x) = x and g(x)= 3x +6 3

e) find f  g(x)

2 g) evaluate f  g( ) 3

f) find g  f(x)

3 h) evaluate g  f( ) 2

ALGEBRA II

HELPFUL NOTES 1.7 Inverse functions

To find the inverse of a function, switch the ______________________.

Equations: 1. Write y instead of f(x) 2. Switch y and x 3. Solve for the new y: f(x) = 3x – 7

f(x) = 2x + 4

f(x) =

4 x 5

f(x) = x 2

f(x) = x 2 – 1

f(x) =

8 x

f(x) =

12 x−2

f(x) = 4

Graphs: graph each function with its inverse on your calculator (if possible). What can you conclude about the graphs? Sketch the inverses for the following graphs: 4 3 5

-2

Coordinates: just switch y and x f(x) = (2, 0), (3, -7), (-1, 4)

f −1 (x) =

Table: just switch y and x f(x) x y

1 -1

2 0

3 -1

4 -4

f −1 (x) x y