Permutation and Combination

Permutation means when the order is important.

Formula: nPr=n!/(n-r)!

Combination means when the order is not important.

Formula: nCr=n!/(n-r)!r!

Example 1.
In how many ways can 8 CD’s be arranged on a shelf?

Since order is important, you use permutation

Example 2:

In how many ways can a sorority of 20 members select a president, vice president and treasury, assuming that the same person cannot hold more than one office.

Order is important

Permutations and Combinations

Permutation: important nPr= n!/(n-r)!

Combination: not important nCr=n!/(n-r)!r!

Examples:

Student council needs a president, vice president, and secretary.

Use a permutation because it's important.

Student council needs 3 equal officers. Combination because it's not important.

Permutation means when the order is important.

Formula: nPr=n!/(n-r)!

Combination means when the order is not important.

Formula: nCr=n!/(n-r)!r!

Example 1.

In how many ways can a beach with ten members choose 5
different life guards?

10P5

10!/5!=

10x9x8x7x6x5x4x3x2x1=362880/5x4x3x2x1=120
So the final answer is 30240.

Example 2.
A football team has a total of fifty players on the team and the coach needs to make five of them team leaders or captains. How many team leaders can be made if the order of the picking is not considered important?

50C5

50!/45!5!=30510144000

Example 3.

If a teacher had a class of twenty studens and gave five students books, how many ways can the books are given if they are all different?

20P5

20!/15!= 1860480

Permutations and Combinations!

Permutation: when the order is important or different.
nPr=n!/(n-r)!

Combination: when the order is not important or identical.
nCr=n!/(n-r)!r!

______________________________________________________

Ex.1: How many ways can a club choose 3 offices; a president, a vice-president and a secretary, if there are 10 members?:

Since the order is important you would use Permutation

10P3=720 ways.

Ex.2: How many ways can 6 books be given to 6 friends if they are all identical?:

Since the books are identical you would use Combination

6C6=1 way.

Permutations and Combinations

5P2=120

5C2=10


This week we learned about permutations and combinations.  Permutations mean important or considered and combinations mean not important or not considered.  There are two separate formulas for permutations and combinations.  The permutation formula is nPr= n!/(n-r)! and the combination formula is nCr= n!/(n-r)! r!.  Basically all you have to do to solve these types of problems is just to plug it into the formula you are given.  So for the first example I did, the 5 in the n in this problem and the 2 is the r in this problem.  You are given that it is permutation so you know to use that formula.  So you would do 5!/3! and your answer is 120.  For the second example I did, the 5 in this problem is the n and the 2 in the problem is the r.  It is the same numbers, but you have to use the other formula since you are told it is combination.  When you use this formula, you get 5!/3!2! and the equals 10.

Permutations and Combinations!

Permutation: when the order is important or different.
nPr=n!/(n-r)!

Combination: when the order is not important or identical.
nCr=n!/(n-r)!r!

Example 1:
In how many ways can a club with 13 members choose 4 different officers?
13P4
13!/9!=13*12*11*10*9*8*7*6*5*4*3*2*1/9*8*7*6*5*4*3*2*1
both 9*8*7*6*5*4*3*2*1 cancel out, so you are left with
13*12*11*10
=17160
In how many ways can the club choose a 4-person governing council?
13C4
13!/9!4!=13*12*11*10*9*8*7*6*5*4*3*2*1/9*8*7*6*5*4*3*2*1*4*3*2*1
both 9*8*7*6*5*4*3*2*1 cancel out, so you are left with
13*12*11*10/4*3*2*1
=715

Example 2:
A teacher has a collection of 20 true-false questions and wishes to choose 5 of them for a quiz. How many quizzes can be made if the order of the questions is considered important?
20P5
20!/15!=20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1/15*14*13*12*11*10*9*8*7*6*5*4*3*2*1
both 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 cancel out, so you are left with
20*19*18*17*16
=1860480
Considered unimportant?
20C5
20!/15!5!
=20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1/15*14*13*12*11*10*9*8*7*6*5*4*3*2*1*5*4*3*2*1
both 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 cancel out, so you are left with
20*19*18*17*16/5*4*3*2*1
=15504

Example 3:
Suppose you bought 4 books and gave one to each of 4 friends. In how many ways can the books be given if they are all different?
4P4
4!/0!
4*3*2*1/1
(((0!=1)))
=24
If they are all identical?
4C4
4!0!4!=4*3*2*1/1*4*3*2*1
both 4*3*2*1 cancel out, so
=1

Venn Diagram

An upside down U, is called the intersection (middle).

A U, is called the union, means everything in the 3 sections.

A with line on top means everything but A.

B with a line means everything but B.

A includes the middle section and A section.

B includes the middle section and B section.

A upside down U B, with a line on top means everything but the intersection.
A U B with a line on top of everything means just the number in the bottom left corner.
A U B line on top of B means everything but the B.

A upside down U B line on top of B means just A.

A upside down U B line of top of A means just B.

A U B line on top of A means everything but B.

Example:

From a survey of a 100 college students, 75 students owned stereos, 45 owned cars, and 35 owned stereos and cars.

How many students owned neither?

To find the answer, you have to subtract the total amount of students (100), from the number of students with a car,or a stereo (85). The answer is, 15 own neither.

Permutation and Combination

Permutation means arrangement of things while Combination means selection of things. Permutation is used when the order of things is important and combination is used when the order of things has no importance.

Permutation(important) - nPr = n!/(n-r)! n - bigger number r - smaller number

Combination(unimportant) - nCr = n!/(n-r)!r!

Ex 1. A coach must choose five starters from a team of 12 players. How many different ways can the coach choose the starters?

The order is not important so 12C9 = 12!/3!9! = 792

Ex 2. In how many ways can a sorority of 20 members select a president, vice president and treasury, assuming that the same person cannot hold more than one office.

The order is important so 20P3 = 20!/17! = 6840

Permutation and Combination

Permutation and combination are ways to solve problems that ask about the order of things. Most of the will ask in how many ways are possible if oder is not important or is important. These problems are just problems of order involving choices. If you follow the formula you will almsot always likely get the right answer.

Permutation: where the order of selection is important
Combination: where the order of the selction in not important

Formulas:
Permutation- n!/(n-r)!
Combination- n!/(n-r)! r!

*There is also a type of permutation with repetition. It means the order is importatn but you may have of or more of a type of item. When you have a problem like this you use the formula:
n!/n1! n2! n3!........

EX1 5P2
means 5 Permutation of 2
A. plug into formula and get 5!/3!
you get 20

EX2 There are 2 spots in a dance. A teacher will pick form her 6 students to perform that number. List all possible ways the spots can be filled.
A. It is combination because they have no specifics about the spots.
B. Use formula and plug in 6!/4! 2! = 10

Venn Diagrams

1) When you have A U B- you only use the middle cause thats the intersection
2) A upside down U B- union
3) A (line on top)- everthing but A
4) B (line on top)- everything but B
5) A- A and the intersection
6) B- B and the intersection
7) A upside down U B (line over everything)- everything but the intersection
8) A U B (line over everything)- number in the conner
9) A U B (line on top of b)- everything but the B
10) A upside down U B (line on top of B)- just A
11) A upside down U B (line of top of A)- just B
12) A U B (line on top of A)- everything but B

Example:
50 people went to the beach, 25 got bitten by mosquitoes and 12 got sunburned and 7 got both. how many got none?
Answer: 20!

Permutation land Combination

Sooooooooooo this week we learned Coach Matt taught us about permutation and combination:

-With permutation the order is important. Formula: nPr = n!/(n-r)!
-With combination the order isn't important. Formula: nCr = n!/(n-r)!r!

Example 1: How many 2 digit numbers can you make using digits 1, 2, 3, and 4 withour repeating the digits?

Since you don't want to repeat the order is important so you need to use permutation. 4P2 = 4!/(4-2)! = 12

Example 2: We need to form a 5 a side team in a class of 12 students. How many diffent teams can be formed?

Since the order isn't important you use combination. 12C5 = 12!/(12-5)!5! = 792

Example 3: How many 6 letter words can we make using the word LIBERTY without repitition?

7P6 = 7!/(7-6)! = 5040

Example 4: In how many ways can you make a committee of 3 students out of 10 students?

10C3 = 10!/(10-3)!3! = 120

--Danielle.

The multiplication principle

Multiplication Principle.

The multiplication principle is very simple. It basically states that if you have an action that can be performed in 1 way, and for each of these ways another action can be performed in 2 ways, then the two actions can be performed together in X1X2 ways.
Basically, you just multiply the two or more numbers together to find out how many ways something can be done. This method can only be used though if you dont have restrictions.

Ways to help the problem to be easier to solve or work out is to draw a venn diagram.

-Restriction:A set of rules of what you cant use or do to find your set of how many ways.

For example, if a number can't be repeated in the problem.

Ex1 :
There are 10 runners in a race. They can win first, second, or third place. In how many ways can these runners win each award?

You would solve this problem by first:

-making sure you know how many places there are to win a prize.

-make sure that you know that a runner can't be repeated.

(once a prize is given away there is one less runner.)

So you would solve this problem by multiplying 10 x 9 x 8 which equals 720 different ways.

Venn Diagrams

1.) A upside down U B=middle and its called the intersection
2.) A U B= everything in the 3 sections and its called the union
3.) A with line= everthing but A
4.) B with line= everthing but B
5.) A= middle section and A section
6.) B= middle section and B section
7.) A upside down U B, with a line on top- everything but the intersection
8.) A U B with a line on top of everything= just the number in the bottow left corner
9.) A U B line on top of B= everything but the B
10.) A upside down U B line on top of B= just A
11.) A upside down U B line of top of A= just B
12.) A U B line on top of A= everything but B

http://www.youtube.com/watch?v=gaEvsvHb4kk

Multiplication Principal :)

If an action can be performed in X1 ways, and for each of these ways another action can be performed in X2 ways, then the two actions can be performed together in X1X2 ways. This is only if you don't have restrictions.

Restriction:A set of rules of what you cant use or do to find your set of how many ways.


Examples:
1.Kasy has 3 white shirts, 5 jackets, and 4 jeans. How many different outfits would she be able to wear?

In order to find out the correct answer you have to mulitiply three,five,and four all together and that will give you the total amount of outfits.
3x5x4=60 outfits



2.Mike is hungry and wants to make a sandwich. They have turkey,smoked ham,or baloney to choose from and he can have mustard,mayo,or ketchup with wheat,white,or brown wheat bread. How many different way can he make his sandwhich?

In order to solve this you will have to multiply the total amount of meat,components,and breads all together.
3x3x3=27 ways

Venn Diagrams

Venn Diagram Rules:

A upside down U B => intersection, or middle of the diagram

A U B => union, or total value of 3 sections

line over A => compliment of A, or everything but A

line over B => same thing but with B

A => intersection and A

B => intersection and B

Compliment of A intersection B => everything but the intersection

Compliment of A U B => everything that's not in the circles

A U compliment of B => everything but B

A intersection of B, compliment of B => only A

A intersection of B, compliment of A => only B

A U B compliment of A => everything but B

An upside down U, which is called the intersection, means the middle. A U, which is called the

union, means everything in the 3 sections. A with line on top means everything but A. B with a

line means everything but B. A includes the middle section and A section. B includes the

middle section and B section. A upside down U B, with a line on top means everything but the

intersection. A U B with a line on top of everything means just the number in the bottom left

corner. A U B line on top of B means everything but the B. A upside down U B line on top of B

means just A. A upside down U B line of top of A means just B. A U B line on top of A means

everything but B.

Example:

40 people went to the park, 25 were playing games with their dogs and 12 were

riding bike and 7 did both. How many did neither?

And the answer is 10 did neither.

The Multiplication Principle!

If an action can be performed in X1 ways, and for each of these ways another action can be performed in X2 ways, then the two actions can be performed together in X1X2 ways. This is only if you don't have restrictions.

Restriction:A set of rules of what you cant use or do to find your set of how many ways.

_______________________________________________________

Ex.1: There are 7 people in line at a restaurant; how many different ways can their order be taken?

You would multiply 7*6*5*4*3*2*1, and the answer would be how many ways.

This can also be expressed as 7!.

7!=5040

Ex.2: If Jimmy has 3 different shirts and 5 different pants, how many combinations does he have?

You would multiply 3*5 and that would be your answer.

Jimmy has 15 different shirt/pants combinations.

Venn Diagrams!:)-Kayla

1.) A upside down U B=middle and its called the intersection
2.) A U B= everything in the 3 sections and its called the union
3.) A with line= everthing but A
4.) B with line= everthing but B
5.) A= middle section and A section
6.) B= middle section and B section
7.) A upside down U B, with a line on top- everything but the intersection
8.) A U B with a line on top of everything= just the number in the bottow left corner
9.) A U B line on top of B= everything but the B
10.) A upside down U B line on top of B= just A
11.) A upside down U B line of top of A= just B
12.) A U B line on top of A= everything but B

Example:
40 people went to the beach, 25 got bitten by mosquitoes and 12 got sunburned and 7 got both. how many got none?
Answer: 10! :)

Venn Diagrams

Okay I'm not technology advanced enough to know how to draw a venn diagram on here, so you are just going to have to imagine.
Circle A has 410 in it, circle B has 100 in it, and the over lap circle has 40 inside.  Inside the rectangle the number is 450, and all together the number is 1,000. 

A and B in this problem represent blood types.  So for circle A you have 410 people with blood type A and for circle B you have 100 people with blood type B.  40 which is the number inside the circle is the number for blood type A and blood type B.  The number 450 which is the inside left corner of the rectangle represents all the people out of the 1,000 people that have neither blood type A or B.  So for number one they ask you for A upside down u of B  and you get 40.  That is the number in the middle of the circle.  For number two you have A UB which is 550-everything in the circle added together.  Number three you have A with a line on top.  This means you add the B and your 450 together to get 550.  Number four is B with the line on top and you get 860 from adding A and 450 together.  A with a line on tope with an upside down u B means just B, which is 100.  A with an upside down u B with a line on top means just A, which is 410. 

Multiplication Priciple

The Multiplication Principle:
If an action can be performed in X1 ways, and for each of these ways another action can be performed in X2 ways, then the two actions can be performed together in X1X2 ways.
(In other words you just multiply the two or more numbers together to find out how many ways something can be done. This however is only if you do not have restrictions.)

*It helps to draw diagrams for finding how many ways especially if you have restrictions.

-Restriction:A set of rules of what you cant used or do to find your set of how many ways.

Ex1
There are 6 people sitting in chairs at a play. In how many ways can these people be arranged?
A. You should draw out six places to get an idea then draw numbers 1-6 one in each place. Notice that your restriction is that once person cannot be used twice. You do this because once a person is seated the rest can be filled by the the other people.
B.Then mulitiply all numbers in each place 1*2*3*4*5*6 =720 (this can also be expressed 6!)
C.So they're 720 ways these 6 people can be seated.

Venn Diagrams!

Helllllo, I hope everyone had a great weekend!















*A upside down U B=the intersection of A&B, which would be the number in the middle

*AUB=the union, which is everything

*a line over A=the compliment of A, which is everything but A

*a line over B=the compliment of B, which is everything but B

*a line over A, upside down U,B=the intersection of A&B with the compliment of A, which would be B

*A, upside down U, line over M=the intersection of A&B with the compliment of B, which would be A

*AUB with a line over the whole thing=the compliment of the union, which would be everything but what is inside of the circles

*AUB with a line over B=the union with the compliment of B, which is everything but B




Example 1:

Circle A=20

Circle B=22

Middle=8

Neither=50

Total=100

Find each of the following:

A upside down B (intersection of A&B)= 8

AUB (union, which is everything)= 50

Line over A (compliment of A, which is everything but A)= 72

Line over B (compliment of B, which is everything but B)= 70
A upside down B with a line over everything (compliment of the intersection, which is everything but the middle)= 92

AUB with a line over everything (compliment of the union, which is everything outside of the circles)= 50

A upside down U with a line over B (intersection of A&B with the compliment of B, which is A)= 20

AUB with a line over B (the union of A&B with the compliment of B, which is everything but B)= 70

****SORRY FOR ALL THE SPACES.. DON'T KNOW HOW TO FIX THEM.

Chapter 15 Venn Diagrams

Hope everyone is enjoying their weekend! Also I hope your having a good trip Brob, good luck to Jacob :)


S0000 since Brob isn't here we have Coach Matt and this week he taught us about Venn Diagrams:




1) When you have A U B- you only use the middle cause thats the intersection
2) A upside down U B- is the union
3) A with a line on top- everthing but A
4) B with a line on top- everthing but B
5) Just A- means A and the intersection
6) Just B- means B and the intersection
7) A upside down U B, with a line on top- everything but the intersection
8) A U B, with a line on top- just the number in the conner
9) A U B, line on top of B- everything but the B
10) A upside down U B, line on top of B- just A
11) A upside down U B, line of top of A- just B
12) A U B, line on top of A- everything but B




Eample 1:


1. A upside down U B = 40
2. A U B = 550
3. A with a line on top = 550
4. B with a line on top = 860
5. A upside down U B, A with a line on top = 100
6. A upside down U B, B with a line on top = 410
7. A U B, with a line on top of everything = 450
8. A U B, B with a line on top = 900

---Danielleeee
This is prob gonna be really spaced out cause i have pictures so sorry..

Function Notation

(f+g)(x) means add the equations

(f-g)(x) means subtract the equations

(f *g)(x) means multiply the equations

(f/g)(x) means to divide the equations

(f ᵒ g)(x) or f(g(x)) means to replace all of the x's in the equation with the g(x)-(f ᵒ g)

If x is replaced with a number, plug it into the equation.

Example 1
Determine if f(x)=3x^3-2x^2-4x-5 is even, odd, or neither.
f(-x)=3(-x)^3-2(-x)^2-4(-x)-5
=3x^3-2x^2-4x-5
Even

Domain and Range

DOMAIN AND RANGE

Domain-the interval of x values where the graph exists.
-Range-the interval of y values where the graph exists.
-Zeros x-int root-set=0 solve for x.
-To be a function the graph must pass the vertical line test.
-If given points the domain is the list of all x-values and the range is the list of all y-values in { }.
To find the Domain and Range you..
1. polynomials: domain (-infinity, infinity) always range (-infinity, infinity) odd always
2. square root: y=square root of x+/-number +/-a

To find domain of square root:
1. set inside the root=0 and solve
2. set up intervals
3. plug in, if you get a negative x is the interval
4. write answers in interval notation-use [ by the number range [a,infinity)

square root: y=square root of number - x^2 +/-a
To find domain:
1. set inside =0 solve
2. put answers in [-, ]
range: [0,square root of number] if a=0.
[0+a,square root of number+a] if a is positive
[0-a,square root of number-a] if a is negative

fraction:
To find domain:
1. factor top and bottom
2. cancel if possible and mark the number canceled
3. set bottom =0 and solve for x-values
4. write in interval notation stopping at number's found in 2 and 3

absolute value: domain: (-infinity,infinity)
range: [0+a,infinity) if opens up and a is positive
[0-a,infinity) if opens up and a is negative
(-infinity,0-a] if opens down and a is negative
(-infinity,0+a] if opens down and a is positive

Example 1:
Find the domain and range and tell whether each is a function.
g(t)= t+2/t^2+5t+8
1. t+2/(t+4)(t+2)
2. the (t+2)'s cancel x=-2
3. t+4=0 t=-4
4. (-infinity,-4) u (-4,-2) u (-2,infinity)

Function Notation

(f + g)(x) => add

(f - g)(x) => subtract

(f x g)(x) => multiply

(f/g)(x) => divide

(f o g)(x) => replace x's in f(x) with a # then plug answer into g(x)

if x is replaced with a # plug it into the equation.

That's how it's done.

function notation

lets talk bout function notation real quick

NOTES

(f+g)(x)-add equations

(f-g)(x)-subtract equations

(f.g)(x)- multiply equations

(f/g)(x)- divide

(fog)(x) or f(g(x))-replace all x's in the f(x) equation with the equation g(x)--If x is replaced with a number plug into the equation


Let f(x)= x+ 4 and g(x)= x + 6

Ex: Find (f + g)(x)

(x +4) + (x + 6)=

2x + 10

Ex: Find (f-g)(x)

(x + 4) - (x + 6)=

-2

-Domain-the interval of x values where the graph exists.
-Range-the interval of y values where the graph exists.
-Zeros x-int root-set=0 solve for x.
-To be a function the graph must pass the vertical line test.
-If given points the domain is the list of all x-values and the range is the list of all y-values in { }.

To find the Domain and Range you..

1. polynomials: domain (-infinity, infinity) always range (-infinity, infinity) odd always

2. square root: y=square root of x+/-number +/-a

To find domain of square root:

1. set inside the root=0 and solve

2. set up intervals

3. plug in, if you get a negative x is the interval

4. write answers in interval notation-use [ by the number range [a,infinity)

square root: y=square root of number - x^2 +/-a

To find domain:

1. set inside =0 solve

2. put answers in [-, ]

range: [0,square root of number] if a=0.

[0+a,square root of number+a] if a is positive

[0-a,square root of number-a] if a is negative

fraction:

To find domain:

1. factor top and bottom

2. cancel if possible and mark the number cancelled

3. set bottom =0 and solve for x-values

4. write in interval notation stopping at number's found in 2 and 3

absolute value: domain: (-infinity,infinity)

range: [0+a,infinity) if opens up and a is positive

[0-a,infinity) if opens up and a is negative

(-infinity,0-a] if opens down and a is negative

(-infinity,0+a] if opens down and a is positive

Example 1:

Find the domain and range and tell whether each is a funtion.

g(t)= t+2/t^2+5t+6

1. t+2/(t+3)(t+2)

2. the (t+2)'s cancel x=-2

3. t+3=0 t=-3

4. (-infinity,-3) u (-3,-2) u (-2,infinity)

Funtion Notation!

Hopee everyone had a good four day weekend :)

Have a good trip Brob, be safeee!

So this week one of the things we learned about was function notation:

(f+g)(x)--> add equations
(f-g)(x)--> subtract equations
(f.g)(x)--> multiply equations
(f/g)(x)--> divide
(fog)(x) or f(g(x))--> replace all x's in the f(x) equation with the equation g(x)
--If x is replaced with a number plug into the equation


Example 1:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(f-g)(x)
(x^2+x)-(x+1)
x(x+1)-(x+1)

Example 2:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(f/g)(x)
x^2+x/x+1=x(x+1)/(x+1)
**The x+1's cancel, you are left with x
=x

Example 3:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(fog)(x)
(x+1)^2+x+1
x^2+2+x+1
x^2+x+3

----Daniellleeee

Function Notation!

The formulas:

~(f+g)(x)--> add equations
~(f-g)(x)--> subtract equations
~(f*g)(x)--> multiply equations
~(f/g)(x)--> divide
~(fog)(x) or f(g(x))--> replace all x's in the f(x) equation with the equation g(x)
--If x is replaced with a number plug into the equation.

Ex1. If f(x)=x+1 and g(x)=x^2-1 Find the rule for (f+g)(x):

(x+1)+(x^2-1)= x(x+1)

Ex2. If f(x)= 1/x and g(x)=x+1 find (f o g)(x)or f(g(x)):

f(g(x))= 1/x+1

Ex3. If f(x)=x+1 and g(x)=x^2-1 find (f/g)(x):

x+1/x^2-1= 1/x-1

Function Notation

Function Notation is ways of writing and solving algebraic equations. The equations can be short or long and contain none to infinite variables. There are five different types of function notations.

*To solve a function notation all you do is identify the function, find what you must do to solve or simplify the function, plug in and actually simplify or solve.

*Most mistakes are made with pluging in for (f 0 g) and (g 0 f) and simpily solving.

*Types of functions:
-(f+g) (x) = add equations
-(f-g) (x) = subract equations
-(f *g) (x) = multiply equations (closed circle)
-(f/g) (x) = divide equations
-(f 0 g) (x) or (g 0 f) (x) = replace all x's in the f(x) equation with the equation g(x). Or replace all x's in the g(x) equation with the equation g(x)

*If (x) is replaces with a # plug into equation Ex:f(2) ; (f+g) (2)

Ex1. f(x) = x+1 g(x) = 1-x Find rule for:
(f+g) (x) (f 0 g) (x)
rule states add equations rule states replace all x's in f(x) with g(x)
(x+1) + (x-1) = 1 (1-x)+x= 1

Function Notation

Helllllo everyone :)
Hope you all have a wonderful 4 day weekend!

(f+g)(x)--> add equations
(f-g)(x)--> subtract equations
(f.g)(x)--> multiply equations
(f/g)(x)--> divide
(fog)(x) or f(g(x))--> replace all x's in the f(x) equation with the equation g(x)
--If x is replaced with a number plug into the equation

Example 1:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(f+g)(x)
(x^2+x)+(x+1)
x(x+1)+(x+1)
x(x+1)^2

Example 2:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(f-g)(x)
(x^2+x)-(x+1)
x(x+1)-(x+1)

Example 3:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(f.g)(x)
(x^2+x)(x+1)
x^3+x^2+x^2+1
x^4+x^3+1

Example 4:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(f/g)(x)
x^2+x/x+1=x(x+1)/(x+1)
**The x+1's cancel, you are left with x
=x

Example 5:
Let f(x)=x^2+x and g(x)=x+1. Find each of the following.
(fog)(x)
(x+1)^2+x+1
x^2+2+x+1
x^2+x+3

Example 6:
Find each of the following.
a. (f+g)(0)
f(x)=square root of 25-x^2= square root of 25-0^2 +absolute value of 0=5
g(x)=absolute value of x
b. (f+g)(3)
f(x)=square root of 25-x^2=square root of 25-3^2 +absolute value of 3= 4+3
4+3=7 4-3=1

Determinates

1      2   3       1    2
2      4   6        2   4
17  18  19      17  18

76+204+108-76-108-204=0

First of all, there are different types of maxtrices.  There can be a 2*2, 3*3, 4*4, and many more.  There is different ways to find the determinates of each also.  For a 2*2, all you have to do is multiply diagonally and then subtract them.  For a 3*3, which is what we have here, you do something completely different than you would with a 2*2 matrice.  For 3*3 matrices, you have to recopy the first two columns.  Once you do that, you have to multiply diagonals.  You have to add the diagonals that are going right and you subtract the diagonals that go left.  Once you do all of the stuff you just add and subtract all the numbers together and you have your answer.  For this problem we are given the answer I calculated was zero.  All the numbers canceled out, which makes it easier to solve!

Inverses

To have an inverse that is a function it must pass the horizontal line test.

To find the inverse you switch the x and y and solve for y.

To prove something is an inverse do (f ᵒ f^-1)(x) and (f ^-1 ᵒ f)(y). Both should simplify to x.

F^-1(x) means inverse.

Ex 1. f(x) = x – ½ and g(x) = 2x + 1 Show that f(x) and g(x) are inverses

(f ᵒ g)(x) = f(g(x)) = 2x + 1 – 1/2 = x

(g ᵒ f)(x) = g(f(x)) = 2(x – 1/2) + 1 = x

Ex 2. Find the inverse of f(x) = 2x – 3

x = 2y – 3 y = x + 3/2

Ex 3. Find the inverse of y = sqrtx – 4

x^2 = sqrty + 4^2 x^2 = y – 4 y = x^2 – 4

Domain and Range

Domain is the interval of x values where the graph exists. Range is the interval of y values where the graph exists. In order to be a function the graph must pass the vertical line test. You use () by ∞ or an open circle. You use [] by the real numbers. #1 polynomials X- Domain: (-∞,∞)-> always! Range: (-∞,∞)-> always! X²- Domain: (-b/2a,∞)-> if it opens upwards Domain: (∞,-b,2a)-> if it opens downwards #2 square roots Domain: 1 set inside the root equal to zero 2 solve 3 plugin (if you get a negative, the interval will not work) 4 write in interval notation Range: [a,∞) #3 square roots Domain: 1 set aside equal to zero 2 solve 3 put the answer in [#,#] Range: If a=0, [0,#²] If a is positive, [0+a,#²+a] If a is negative, [0-a,#²-a] #4 fractions Domain: 1 factor the top and bottom 2 cancel of possible 3 set bottom equal to zero 4 solve x value from step 3 5 write in interval notation, stopping at numbers found in 2 and 3 #5 absolute value Domain: (-∞,∞) Range: If positive or goes upwards, [0+a,∞) If negative or goes downwards, [0-a,∞) Example 1 y = √(x + 4) Domain: [-4,∞) Range: [0,∞) Example 2

D and R

Domain and Range



-Domain-the interval of x values where the graph exists.
-Range-the interval of y values where the graph exists.
-Zeros x-int root-set=0 solve for x.
-To be a function the graph must pass the vertical line test.
-If given points the domain is the list of all x-values and the range is the list of all y-values in { }.

The way to find Domain and range

1. polynomials: domain (-infinity, infinity) always

range (-infinity, infinity) odd always


2. square root: y=square root of x+/-number +/-a


to find square root of domain

1. set inside the root=0 and solve

2. set up intervals

3. plug in, if you get a negative x is the interval

4. write answers in interval notation-use [ by the number


range [a,infinity)



3. square root: y=square root of number - x^2 +/-a


To find domain:

1. set inside =0 solve

2. put answers in [-, ]


range: [0,square root of number] if a=0.


[0+a,square root of number+a] if a is positive


[0-a,square root of number-a] if a is negative



4. fraction:


To find domain:

1. factor top and bottom

2. cancel if possible and mark the number cancelled

3. set bottom =0 and solve for x-values

4. write in interval notation stopping at number's found in 2 and 3



5. absolute value: domain: (-infinity,infinity)


range: [0+a,infinity) if opens up and a is positive


[0-a,infinity) if opens up and a is negative


(-infinity,0-a] if opens down and a is negative


(-infinity,0+a] if opens down and a is positive

Find domain and range


g(t)= t+2/t^2+5t+8


1. t+2/(t+4)(t+2)


2. the (t+2)'s cancel x=-2


3. t+4=0 t=-4


4. (-infinity,-4) u (-4,-2) u (-2,infinity)

Function notation/inverses

Function Notation

-(f+g)(x) ---> add equations

-(f-g)(x) ---> subtract equations

-(f closed circle g)(x) ---> multiply (only if closed circle)

-(f/g)(x) ---> divide

-(f open circle g)(x) or f(g(x))---> replace all x's in the f(x) equation with the equation g(x).

-If x is replaced w/ a number, plug into the equation. Ex: f(2); (f+g)(2); etc.

Example: x+2/ (x^2-4) = x+2/(x+2)(x-2)

The x+2's then cancel out, leaving you with 1/x-2

Inverses

To have an inverse that is a function it must pass the horizontal line test.

-To find the inverse you switch the x+y and solve for y.

-To prove something is an inverse do (f open circle f^-1)(x) and (f^-1 open circle f)(x). Both should simplify to x.

-f^-1(x) means inverse.

Example: Find the inverse of y= square root of x+3

x^2=square root of y+3^2

-square root ^2 cancels out

You're then left with x^2=y+3

-subtract 3 from each side.

Your answer is y=x^2-3 or f^-1(x)=x^2-3

WOOOOAhhhhh

FUNCTIONS:

if you have the formula (f + g)(x) it means to add the equations

if you have the formula (f – g)(x) it means to subtract the equations

if you have the formula (f ●g)(x) it means to multiply the equations

if you have the formula (f/g)(x) it means to divide the equations

(f ᵒ g)(x) Or f(g(x)) means to replace all x’s in the f(x) equation with the equation g(x)-fog

You should factor your final answer. If x is replaced with a number, plug it into the equation.

Example

Determine if f(x)= 2x^3 - 3x^2 - 4x - 4 is even, odd, or neither

1) I'll plug in (-x) in for x, and simplify

2) f(-x)= 2(-x)^3 - 3(-x)^2 - 4(-x) - 4

= -2x^3 - 3x^2 + 4x + 4

3) This is neither what I started with, nor the exact opposite of what I started with

4) The answer is neither

Function Notations:

(f + g)(x) Means to add
equations

(f – g)(x) Means to subtract
equations

(f ●g)(x) Means to multiply
equations

(f/g)(x) Means to divide
equations

(f ᵒ g)(x) Or f(g(x)) – replace all x’s in the f(x)
equation with the equation g(x)-fog just means f of x

You should factor your
final answer. If x is replaced with a number, plug it into the equation.

Example one:
(f/g)(x)
2x+3/4x²-9

The denominator factors to (2x-3)(2x+3) and the (2x+3) cancels
out which leaves you with 1/2x-3.

Example two: (f-g)(x)
2x+3 - ( 4x²-9)

which foils to -4x²+2x+12, then you take a -1 which leaves you with
4x²-2x-12 l then you factors of 48 that subtract to give you -2 which is -8 and
6, 4x²-8x+6x-12, you then take out a 4x
and 6 which gives you 4x(x-2)+6(x-2). You
are then left with (4x+6)(x-2).

inverses

Notes
- to have an inverse, it must pass horizontal line test to be a function
- you switch x and y and solve for y

Lets do an example homie

Ex: Find the inverse of y=8x-4

This is a line which does pass the horiontal line test.
Be sure to check that before you waste uour time on a problem

x=8y-4

8y=4

y=1/2

To write something as an inverse, you use f^-1

This final answer would be f^-1= 1/2

There it go ya heard me. Rico Gathers all caught up on his blogs ya digg.

christmas break blogs

lemme talk to you real quick about sectors of circles.

Formulas: k=1/2r^2(theta) s=r(theta)
or
k=1/2 rs

k=area of a sector
r= radius
theta= central angle
s=arc length

Not much explaining to do besides juat plug into the formula

Ex: s sector of a circle has arc lenth 4cm and are 20 cm^2. Find radius and central angle.

s=4
k=20
r=?
theta=?

r=2k/s= 2(20)/4 r=10

4=10(theta) theta= 5/2

there it is boi. Ricky G done with the christmas blogs

christmas break blogs

Since this is still blogs from christmas break, now im gonna do coterminal angles

Coterminal angles are like non-reduced fractions.
a.) degrees + 360 degrees= coterminal angle

Ex: Find a positive coterminal angle to -30 degrees.

-30 + 360 = 330 degrees.
Yes it is that easy

b.) radians + 2pi n= coterminal angle

Ex: Find a coterminal angle to pi/3

pi/3 + 2pi = 7/3 pi

dat was pretty simple fa sho

christmas break blogs

its ya boy rico gathers tryin catch up on his blogs real quick. since we can do these blogs on anything im gonna do this one on degrees and radians.

To covert from degrees to radians you have to do "degrees x pi/180"

EX: Convert 45 degrees to radians

45 x pi/180 =

1/4 x pi=

pi/4

to convert from radians to degrees you do radians x 180/ pi

Ex: Convert pi/3 to degrees

pi/3 x 180/ pi

pi's cancel out

60 degrees.

Yea dats wats up

Inverses

Find the inverse of f(x)=2x-3
x=2y-3
y=x+3/2

To have an inverse that is a function it must pass the horizontal line test.  The horizontal line test is when you draw a line across your paper and if your equation that you are given hits the line more than one time, your equation can not have an inverse.  If the line only crosses your equation one time, then it can have an inverse and you can solve the equation.  To find the inverse you switch the x and y in your equation given and then you solve for y.  So for this equation I am given, I used the horizontal line test and since this equation is linear, it will pass the horizontal line test because it only crosses once.  To solve this equation, I switched the x and y and then I added 3 to both sides and got 2y=x+3.  Then I divided everything by 2 and I got y=x+3/2.  So the inverse of this function is y=x+3/2.

Function Notations

This week one of the many things we learned was function notations…

Function Notations:

(f + g)(x) - add equations

(f – g)(x) - subtract equations

(f ●g)(x) - multiply equations

(f/g)(x) - divide equations

(f ᵒ g)(x) or f(g(x)) – replace all x’s in the f(x) equation with the equation g(x)
*fog just means f of x J

You should factor your final answer. If x is replaced with a number, plug it into the equation.

Example 1: (f-g)(x)
2x+3 - ( 4x²-9)
-4x²+2x+12
4x²-2x-12
4x²-8x+6x-12
4x(x-2)+6(x-2)

(4x+6)(x-2)

Example 2: (f/g)(x)
2x+3/4x²-9
You can factor the denominator:
2x+3/(2x-3)(2x+3) and cancel:

1/2x-3

Hope everyone had a gooood weekend!

----Danielleeeee

Domain and Range :)

-Domain-the interval of x values where the graph exists.
-Range-the interval of y values where the graph exists.
-Zeros x-int root-set=0 solve for x.
-To be a function the graph must pass the vertical line test.
-If given points the domain is the list of all x-values and the range is the list of all y-values in { }.

To find the Domain and Range you..

1. polynomials: domain (-infinity, infinity) always

range (-infinity, infinity) odd always

2. square root: y=square root of x+/-number +/-a

To find domain of square root:

1. set inside the root=0 and solve

2. set up intervals

3. plug in, if you get a negative x is the interval

4. write answers in interval notation-use [ by the number

range [a,infinity)

3. square root: y=square root of number - x^2 +/-a

To find domain:

1. set inside =0 solve

2. put answers in [-, ]

range: [0,square root of number] if a=0.

[0+a,square root of number+a] if a is positive

[0-a,square root of number-a] if a is negative

4. fraction:

To find domain:

1. factor top and bottom

2. cancel if possible and mark the number cancelled

3. set bottom =0 and solve for x-values

4. write in interval notation stopping at number's found in 2 and 3

5. absolute value: domain: (-infinity,infinity)

range: [0+a,infinity) if opens up and a is positive

[0-a,infinity) if opens up and a is negative

(-infinity,0-a] if opens down and a is negative

(-infinity,0+a] if opens down and a is positive

Example 1:

Find the domain and range and tell whether each is a funtion.

g(t)= t+2/t^2+5t+6

1. t+2/(t+3)(t+2)

2. the (t+2)'s cancel x=-2

3. t+3=0 t=-3

4. (-infinity,-3) u (-3,-2) u (-2,infinity)

--Kaylaaaaa