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

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

the inverse of a function is defined by

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

**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

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