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

Composition of Functions Inverse Functions

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

Introduction Notation for composition of functions: Alternate notation:

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) ) 

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

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.

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

Now You Try!!!

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)

How to find the inverse

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.

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

Try It! Find the inverse