In Class 1.4, Operations and Composition of Functions€¦ · In Class 1.4, Operations with...

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In Class 1.4, Operations with Functions, Composition of Functions Name: Algebra II, Unit 1: Functions We can perform operations on functions just like we perform operations on numbers or expressions. Note that the domain of the new function may be different than the domain of () (). Example: Given, = 5 6 = ! 5 + 6, determine a) + ( + ) () b) () Given, = + 2 = ! 4 c) () d) !(!) !(!) ! ! () Composition of functions: When going shopping at a store in the mall, you saw an item that was worth $100. However, that item is on sale for 20% off, and also every item in that store will have another $20 off. If you want to buy that item, would you like the cashier to take off $20 first , then 20% or take off 20% first , and then $20 ? Do you care? Think of the following functions: () = 20% = 0 .80 () = 20 A composition of functions is like a series of machines, for example: If = ! 3 and = 2 5, then ( 4 ) looks like this: Recall the function machine puzzle from In Class 1: In what order should you stack the machines so that when 6 is dropped into the first machine, and all four machines have had their effect, the last machine's output is 11? () = () = ( 2) ! () = 2 ! 7 () = 2 1 1) Solve: 6 2) What should be the next function composition to get to the final output of 11?

Transcript of In Class 1.4, Operations and Composition of Functions€¦ · In Class 1.4, Operations with...

Page 1: In Class 1.4, Operations and Composition of Functions€¦ · In Class 1.4, Operations with Functions, Composition of Functions Name: Algebra II, Unit 1: Functions We can perform

In Class 1.4, Operations with Functions, Composition of Functions Name: Algebra II, Unit 1: Functions

We can perform operations on functions just like we perform operations on numbers or expressions. Note that the domain of the new function may be different than the domain of 𝒇(𝒙) 𝒂𝒏𝒅 𝒈(𝒙).

Example: Given,𝑓 𝑥 = 5𝑥 − 6 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥! − 5𝑥 + 6,determinea) 𝑓 𝑥 + 𝑔 𝑥 𝑜𝑟(𝑓 + 𝑔) (𝑥) b)𝑓 𝑥 – 𝑔 𝑥 𝑜𝑟 𝑓 − 𝑔 (𝑥)

Given,𝑓 𝑥 = 𝑥 + 2 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥! − 4

c) 𝑓 𝑥 𝑔 𝑥 𝑜𝑟 𝑓𝑔 (𝑥) d)!(!)

!(!)𝑜𝑟 !

!(𝑥)

Compositionoffunctions:Whengoingshoppingatastoreinthemall,yousawanitemthatwasworth$100.However,thatitemisonsalefor20%off,andalsoeveryiteminthatstorewillhaveanother$20off.Ifyouwanttobuythatitem,wouldyoulikethecashiertotakeoff$20first,then20%ortakeoff20%first,andthen$20?Doyoucare?Think of the following functions: 𝑓(𝑥) = 𝑥 − 20%𝑥 = 0 .80𝑥

𝑔(𝑥) = 𝑥 − 20 A composition of functions is like a series of machines, for example: If 𝑓 𝑥 = 𝑥! − 3 and 𝑔 𝑥 = 2𝑥 − 5, then 𝑓(𝑔 4 ) looks like this: Recall the function machine puzzle from In Class 1: Inwhatordershouldyoustackthemachinessothatwhen6isdroppedintothefirstmachine,andallfourmachineshavehadtheireffect,thelastmachine'soutputis11?

𝑓(𝑥) = 𝑥 𝑔(𝑥) = −(𝑥 − 2)!

ℎ(𝑥) = 2 ! − 7 𝑘(𝑥) = −𝑥2

– 1

1)Solve:𝑔 𝑘 6

2)Whatshouldbethenextfunctioncompositiontogettothefinaloutputof11?

Page 2: In Class 1.4, Operations and Composition of Functions€¦ · In Class 1.4, Operations with Functions, Composition of Functions Name: Algebra II, Unit 1: Functions We can perform

In Class 1.4, Operations with Functions, Composition of Functions Name: Algebra II, Unit 1: Functions