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:  p(x) = 2x + 1  q(x) = x 2 - 3 Then p ( q(x) ) =  p (x 2 - 3) =  2 (x 2 - 3) + 1 =  2x 2 - 5 Try determining q ( p(x) )

5. Try It Out q ( p(x) ) =  q ( 2x + 1) =  (2x + 1) 2 – 3 =  4x 2 + 4x + 1 – 3 =  4x 2 + 4x - 2

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 graphj(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: x12347 f(x)31427 g(x)72143 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