1. TakeTake out: Pencil, Calculator, Do now sheet
Simplify
Agenda
Do Now
Composition and Inverse Functions
HW: Handout-composition and inverses
Objective: Graph composite functions and compose functions
Determine the inverse of a function

2. Composition of Functions
Inverse Functions

3. Introduction Value fed to first function
Resulting value fed to second function 
End result taken from second function 

4. Introduction Notation for composition of functions:
Alternate notation:

5. Try It Out Given two functions: Then p ( q(x) ) =
p(x) = 2x + 1
q(x) = x2 - 3
Then  p ( q(x) ) =
p (x2 - 3) =
2 (x2 - 3) + 1 =
2x2 - 5
Try determining  q ( p(x) ) 

6. Try It Out q ( p(x) ) = q ( 2x + 1) = (2x + 1)2 – 3 =

7. Decomposition of Functions
Someone once dug up Beethoven's tomb and found him at a table busily erasing stacks of papers with music writing on them.  They asked him ... "What are you doing down here in your grave?"  He responded, "I'm de-composing!!"
But, seriously folks ...
Consider the following function which could be a composition of two different functions.

8. Decomposition of Functions
The function could be decomposed into two functions, k and j

9. Now You Try!!!

10. Does not mean f to the -1 of x, It means inverse of f(x)
Inverse Functions
Inverse Notation
Does not mean f to the -1 of x, It means inverse of f(x)

11. How to find the inverse

12. Verifying Inverses If you can do the composition of the
functions both ways (f(g(x)) and
g(f(x))) and the solution is x for both,
then the functions are inverses.

13. Verifying Inverses Continued
For example:
f(x) = 2x - 4 and g(x) = x/2+ 2
f(f-1(x)) = 2(x/2 + 2) - 4
= x + 4 – 4
= x correct

14. Try It!
Find the inverse