January 25,2009

Test from Unit -5 Functions

Hope guys you are preparing the test and be ready for exam. So we prepared the stuff that we did in class.Hope the test will be fine and tommorrow is the project presentation.So ,we ready to present the project.

ok bye see u tomorrow

Have a nice evening.

Test Reminder

Just to let you guys know that we have a test on Functions on Wednesday. So study the functions booklet we got and you'll be fine.

Hall of Famer of the Semester

I cant' believe that our time in class is slowly coming to an end.
For the last time please vote in the comments area for the Hall of Famer of the Semester.

Hall of Famer of the December/January

Please vote in the comments are for the hall of famer of the December/January.

Polynomial Functions

Hey guys its Deep here

Today in our maths class we did graphs of polynomial functions.Mr.p told us about wabbitemu and infinite algebra and i like it very much.Before doing graphs we must know what is a polynomial function.A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

for all arguments x, where n is a non-negative integer and a₀, a₁,a₂, ..., a_n are constant coefficients.

For example, the function ƒ, taking real numbers to real numbers, defined by

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

So, there are some things to remember .They are as follows:-

The term containing the highest power of x is called the leading term and its coefficent is referred to as the leading coefficent . the power of x contained in the leading term is called the degree of the polynomial.

• Degree 0 – constant
• Degree 1 – linear
• Degree 2 – quadratic
• Degree 3 – cubic
• Degree 4 – quartic
• Degree 5 – quintic
• Degree 6 – sextic (or, less commonly, hexic)
• Degree 7 – septic (or, less commonly, heptic)
• Degree 8 – octic

Functions

Hi
     It's kiran
  Today short peiod because if T.A. so today we did one to one function.                                                  
    Definition
    Function is defined as a relation of x,y where for every x there is one and only one value of y assigend . A one to one funtion has another restriction added on to this : For every y which hass been assigned to the x's no other x has that y (for every y there is one and only one value x)

Examples -:
Question 1: Is function f defined by


f = {(1 , 2),(3 , 4),(5 , 6),(8 , 6),(10 , -1)},

a one to one function?
Solution to Question 1:
  . Two different valuse in the damain namely %and 6 have same output, hence function f is not a one to one funcction.
Question 2: Is function g defined by


g = {(-1 , 2),(0 , 4),(2 , -4),(5 , 6),(10 , 0)},

a one to one function?
Solution to Question 2 :


Consider any two different values in the domain of function g and check that their corresponding output are different. Hence function g is a one to one function

• Question 3: Is  faction f given by
  
  
  
  f(x) = -x ³ + 3 x ² - 2,
  
  a one to one function?
  Solution to Question 3:
  
• A graph and the horizontal line test can help to answer the above question.
  
  
  Since a horizontal line cuts the graph of f at 3 different points, that means that they are at least 3 different inputs x1, x2 and x3 with the same output Y and therefore f is not a one to one function.
• Question 4: Show that all linear functions of the form
  
  
  f(x) = a x + b,
  
  where a and b are real numbers such that a not equal to zero, are one to one functions.
  Solution to Question 4:
  
• We start with f(A) = f(B) and show that this leads to a = b
  
  a(A) + b = a(B) + b
  
• Add -b to both sides of the equation to obtain
  
  a(A) = a(B)
  
• Divide both sides by a since it is not equal to zero to obtain
  
  A = B
  
• Since we have proved that f(A) = f(B) leads to A = B then all linear functions of the form f(x) = a x + b are one-to-one functions.
  
  A = B  

                              Good evening to everyone.

Hi

It's karamveer

Today in the class we had learned about Inverse of the function.

Inverse of Relations.The inverse of the relation is the set of the ordered pairs

obtained by interchanging the coordinates of each ordered pair in the relation.

Inverse of Functions.The inverse of the function f(x) is also a function.It is called an inverse function of f(x) and is written f^-1(x).

Find the inverse of f(x) = 6x-5

To find the inverse of the function follow these steps.

y = 6x-5

Step.1:Replace f(x) by y

y=6x-5

Step.2:Interchange x and y

x=6y-5

Step.3:Solve for y

6y=x+5

y=x+5/2

Step.4:Replace y with f^-1(x)

Graph the function and its inverse

The graphs of f (x) and f^-1(x) are relate to each other in this way:(a,b) lies onthe graph f then the point (b,a) lies on the graph f^-1 and the vice versa.This means that graph of the f function is a reflection of the graph of f^-1 in the line y=x.

Second example I want to show the one how would you determine that f(x) and g(x) are the inverses of each other

Example.1f(x)=2x-5 or

Solution :The function f(x) and g(x) are the inverses of each other if f(g(x))=x and g(f(x))=x.

And...

So that's means.They are the inverses of each other.

Make sure you'll try n doing Example no. 5 on page 14 in the booklet.

Good Evening to everyone.See you all tomorrow@10.30.

Test on Analytic Geometry !

Hey everyone it's Angelique, don't forget to study for our test tomorrow on analytic geometry. I wish you all good luck and good night ! :)

Test Tomorrow

Hi everyone,
Just a reminder that we have a test tomorrow. Don't forget to do your analytic geometry test review. Goodluck!

analytic geometry test !

Hey guys, its Daryl. We have a test tomorrow (Jan.14,2011) (: goodluck guys !

Test Friday January 14th

Hey guys its Chris, just here to remind you all that were are having our last test of the year tomorrow and to not forget to study for this test. we had gotten a review a few days back which should be very helpful towards this test! All the best!!! :)

Reminder!!

Just a reminder that we'll be having a Unit test tomorrow on Analytic Geometry. So study your booklet and the review sheet that Mr. Piatek gave us!! Also reminder that the Principia Mathematica Project is also due tomorrow...so study hard and good luck to all of us..
See you all tomorrow!!!

REMINDER !

The Analytic Geometry test is tomorrow! Good luck to everybody (:
*the precal project is also due tomorrow!

Composition of Functions

Hey everyone,

Today in the class we learned the composition of functions and for the homework we get the Compositions of Functions worksheet.Compositions of functions is represented by (f∘ g)(x).It means the range of f is domain of g.To make it simple wherever you
 see " x" in  function f(x) substitute the function g(x).

Here's one example :

Example.1

f(x)=3x²-5x

g(x)=2x-3

Find (f ∘ g)(x)

Solution:Substitute g(x) wherever you see an x in the function of f(x)

3(2x-3)²-5(2x-3)

Now foil (2x-3)²

3(4x²-12x+9)-5(2x-3)

12x²-36x+27-10x+15

Combine like terms.

12x²-46t+42


Example no. 2

g(a)=a³ -4a²

f(a)=-4a-1

Find (g∘f)(a)

Solution:

Step 1:Where you see "a " in function g(a) substitute the function f(x) .

(-4a-1)³-4(-4a-1)²


Step 2 :Foil it

Formula for (a+b)³ is a³+3a²b+3ab²+b³

(a+b)² is a²+2ab+b²

a=-4a and b = -1


Substitute into the formula


(-4a)³+3(-4a)²(-1)+3(-4a)(-1)²+(-1)³-4((-4a)²+2(-4a)(-1)+(-1)²)
-64a³-48a²-12a-1-4(16a²+8a+1)
-64a³-48a²-12a-1-64a²-32a-4


Step 3 :Combine like terms


-64a³-112a²-44a-5


Just a reminder to everyone the we have a test on Analytic Geometry tomorrow.Make sure you guys do the review for the test that we got for Analytic Geometry.Don't forget  to study.Good Luck to everyone !!!


and...................


One tip for tomorrow's test
Do every solution step by step and use the brackets every time and check for the signs too even if you are very good in handling those signs. It is pretty easy to make mistake in substitution and elimination problems.It is the last test so do your best.


Our Project is due tomorrow.


Have a good evening.See you all tomorrow. 

Analytic Geometry :-Test

Analytic Geometry :-Test

 Just a little reminder to all my friends that we have test on friday from analytic geometry. Mr.p gave review sheet  of that unit. hope u all got all the sums .prepare it well and. do not take stress.  Good luck!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Hey guys its deep here

hiii guys, Lovleen here
yesterday we started new unit 5 functions.Mr.P taught us about operations on functions.He told us how to solve the equations.there has many functions like addition,multiplication,subtraction,division and he told us to solve the sums with the functions.
MR.P gave us worksheet to solve. here is a eg from worksheet solved with the multiplication method:
g(x)=x^2+3x
f(x)=x+4
find (g.f)(x)


= (x^2+3x)(x+4)
= x^3+7x^2+12
last thing, i wanna remind u guys that we have a test on FRIDAY. GOOD LUCK.hope all will do well in test.HAVE A NICE DAY

PRE-CAL TEST (UNIT 4)!

hey guys, its pryanka:)

just a reminder to everyone that we have a test on unit four this FRIDAY (January 14 2011)!!Don't forget to study. If you know how to do all the questions in the review, then you guys will do good on the test! also remember to do the worksheets we got in the past classes which will also help you guys on the test!!! Good- Luck(:

 

Pre cal test on Friday!!

Hey everyone it's Raveen

We have a test this friday on unit 4 so..........

DONT FORGET TO STUDY !

If you guys do the review, the test should be pretty easy

If anyone has any questions I will be glad to help

GOOD LUCK EVERYONE !!!!!!!!!!!!!!!!!!!!!!!!!!!