1. Composition of Functions
Lesson 8.1

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

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

4. 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) ) 

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

6. Using the Calculator
Given
Define these functions on your calculator

7. Using the Calculator Now try the following compositions: g( f(7) )
f( g(3) )
g( f(2) )               
f( g(t) )
g( f(s) )
WHY ??

8. Using the Calculator
Is it also possible to have a composition of the same function?
g( g(3.5) ) = ???

9. Composition Using Graphs
k(x) defined by the graph
j(x) defined by the graph
Do the composition of k( j(x) )

10. Composition Using Graphs
It is easier to see what the function is doing if we look at the values of k(x), j(x), and then k( j(x) ) in tables:

11. Composition Using Graphs
Results of k( j(x) )

12. Composition With Tables
Consider the following tables of values:  x 1 2 3 4 7
f(x)
g(x)
f(g(x)
f(g(1))
g(f(x)
g(f(3))

13. 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.

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

15. Assignment
Lesson 8.1
Page 359
Exercises 1 – 59 odd