function inverse and composition

On Friday continued our new topic of Function. We first did a quick review on how to compose a function and then we learn how decompose it.
If we can compose a function. We must also be able to decompose it.for function decomposition we have to understand the "inner function" and the "Outer function"
for example if we have f(x)= 1-x^2 in order to decompose the function F(x)= x^2 Inner function
g(x)= x+1 Outer function
f(x)=1/x^3
F(x)= x^3 Inner function
g(x)=1/x Outer function
for more information about the function composition visit
http://www.purplemath.com/modules/fcncomp.htm
http://oregonstate.edu/instruct/mth251/cq/FieldGuide/composition/lesson.html
http://www.youtube.com/watch?v=S4AEZElTPDo
After the function decomposition we talked about the function of type
many to one function.
Different from one to one function where every one element of the input or the domain correspond to one element of the output or range. A many to one function is a function where every one element of the input domain correspond to more than one element of the output or range.


many to one function

We have to remember that
Input = Domain which is a set of all the first elements of the ordered pairs of a function
Output = Range which is a set of all the second elements of the ordered pairs of a function.

one to one function

One-to-one function are special because they are invertible

The we learned the horizontal line test which is a test use to determine if a function is a one-to-one. we the horizontal line test when we graph a function
If a horizontal line intersects a function's graph. more than once, then the function is not one-to-one.

Passes the test

Fail the test

Finally we learned the inverse of a function. Considering f(x) the function, the inverse of function f(x) will undo what f(x) did.
if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, if an input x into the function ƒ produces an output y, then inputting y into the inverse function ƒ–(read f inverse) the inverse of a one-to-one function is obtained by switching the role of x and y.ƒ– the inverse of a one-to-one function is obtained by switching the role of x and y.


the inverse of a function is defined by

Not all function has an inverse. those that do are called invertible the one-to-one are special because they are invertible.

General procedure for finding the inverse of a function:

• Interchange the variables - First, we will exchange the variables. We do this because we want to find the function that goes the other way, by mapping the old range onto the old domain. So our new equation is x=2y-5.
• Solve for y -The rest is simply solving for the new y, which gives us:
  2y-5 = x
  2y = x+5
  y = (x+5)/2
  Hence, y^-1(x) = (x+5)/2

logic

After that we finished the geometry unit. Today we started a new unit called logic.We started by looking at what the word "logic" means. we then watched a videos about the word argument, what it means on http://www.youtube.com/watch?v=teMlv3ripSM
We later studied some important words and definitions in logic


Logic: (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. en.wikipedia.org/wiki/Logic

The argument: A fact or statement used to support a proposition; a reason:; A verbal dispute; a quarrel; A process of reasoning; A series of statements organized so that the final statement is a conclusion which is intended to follow logically from the preceding statements, which function as premises... en.wiktionary.org/wiki/argument .
The argument consists of two (2) parts:
1. a set of premises
2. a conclusion

We then started to make a difference between premises and a conclusion, valid argument and invalid argument
A premise is a valid truth
A conclusion is a result of an argument
An argument is called valid if its result is direct consequence of its premises. In a valid argument, if the premises are true, then the conclusion must also be true.
An argument is called invalid when its conclusion is not a direct consequence of it’s premises even if its conclusion are true.
Sound argument: an argument is sound if it is both valid and true

After learning the validity and the invalidity of an argument we learned the induction and deduction.
Induction: when we observe several particular examples of same things, and then conjecture that the “the discovered” pattern must always be true

Deduction: a process of reasoning in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true.

We later did a drag and drop exercise on valid argument and invalid argument, inductive and deductive reasoning.

Mr. K gave us a homework which exercise number 45

The next scribe is Zawadi.