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