Composition of Functions Lesson 8.1. Introduction Value fed to first function Resulting value fed to second function End result taken from second function.

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Presentation transcript:

Composition of Functions Lesson 8.1

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

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

Using the Calculator Given Define these functions on your calculator

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

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

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

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:

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

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))

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

Assignment Lesson 8.1 Page 359 Exercises 1 – 59 odd