Lycée Jean-Piaget - Neuchâtel

Mathematics 2MPES Chapter 2

Compositefunctions

Exercise1You are the foreman at the Sesame Street Number Factory. A huge conveyor belt rolls along, covered with big plastic numbers for our customers. Your two best employees are Kate and Steven. Both of them stand at their stations by the conveyor belt. Steven’s job is: whatever number comes to your station, add 2 and then multiply by 5, and send out the resulting number. Katie is next on the line. Her job is: whatever number comes to you, subtract 10, and send the result down the line to Sesame Street.

a. Fill in the following table.

This number comes down the line: −5 −3 −1 2 4 6 10 𝑥 2𝑥 Steven comes up with this number, and sends it down the line to Katie: −15 −5 5 20 30 40 60 5𝑥 + 10 10𝑥 + 10

Katie then spits out this number: −25 −15 −5 10 20 30 50 5𝑥 10𝑥 b. In a massive downsizing effort, you are going to fire Steven. Kate is going to take over both functions (Steven’s and her own). So you want

to give Kate a number, and she first does Steven’s function, and then her own. But now Kate is overworked, so she comes up with a shortcut: one function she can do, that covers both Steven’s job and her own. What does Kate do to each number you give her? (Answer in words). Kate has simply to multiply by five to cover both Steven’s job and her own job. The algebraic form of Steven’s job is 𝑓 𝑥 =5 𝑥 + 2 . A generalization of this function is 𝑓 𝑥 = 5𝑥 + 10. Chaining this function with Kate’s job, which is 𝑔 𝑥 = 𝑥 − 10 results in the function ℎ 𝑥 = 𝑔 𝑓 𝑥 = 𝑓 𝑥 − 10 = 5 𝑥 + 2 − 10 = 5𝑥 + 10 − 10 = 5𝑥

Conclusion: (Summarize your work by interpreting, transcribing and producing descriptions using your answers above, and the dedicated words of the green post-it.)

A composite function is a function that represents, in one single function, an entire chain of dependent functions. A shortcut to the entire process is realized when you plug the first function into the next one, for instance 𝑔 𝑓 𝑥 . This could also b written as 𝑔 ∘ 𝑓 𝑥

Lycée Jean-Piaget - Neuchâtel

Mathematics 2MPES Chapter 2

Compositefunctions

Exercise2Taylor is driving a motorcycle across the country. Each day he covers 500 miles. A policeman started the same place Taylor did, waited a while, and then took off, hoping to catch some illegal activity. The policeman stops each day exactly five miles behind Taylor.

Let 𝑑 equal the number of days they have been driving. (So after the first day, 𝑑 = 1). Let 𝑇 be the number of miles Taylor has driven. Let 𝑝 equal the number of miles the policeman has driven.

a. After three days, how far has Taylor gone? 500 ∙ 3 = 1,500 miles

b. How far has the policeman gone? 505 ∙ 3 = 1,515 miles

c. Write a function 𝑇(𝑑) that gives the number of miles Taylor has travelled, as a function of how many days he has been traveling. 𝑇 𝑑 = 500𝑑

d. Write a function 𝑝(𝑇) that gives the number of mile the policeman has travelled, as a function of the distance that Taylor has travelled. 𝑝 𝑇 = 𝑇 + !

!""= !"!!

!""= !"!

!""𝑇

e. Now write the composite function 𝑝(𝑇(𝑑)) that gives the number of miles the policeman has travelled, as a function of the number of days he has been traveling. 𝑝 𝑇 𝑑 = !"!

!""∙ 500𝑑 = 505𝑑

Conclusion:

A composite function is a function that represents, in one single function, an entire chain of dependent functions. A shortcut to the entire process is realized when you plug the first function into the next one, for instance 𝑝 𝑇 𝑥 . This could also b written as 𝑝 ∘ 𝑇 𝑥

Lycée Jean-Piaget - Neuchâtel

Mathematics 2MPES Chapter 2

Compositefunctions

Exercise3Rashmi is an honour student by day; but by night, she works as a hit man for the mob. Each month she gets paid $1,000 base, plus an extra $100 for each person she kills. Of course, she gets paid in cash - all $20 bills.

Let 𝑘 equal the number of people Rashmi kills in a given month. Let 𝑚 be the amount of money she is paid that month, in dollars. Let 𝑏 be the number of $20 bills she gets.

a. Write a function 𝑚(𝑘) that tells how much money Rashmi makes, in a given month, as a function of the number of people she kills. 𝑚 𝑘 = 100𝑘 + 1,000

b. Write a function 𝑏(𝑚) that tells how many bills Rashmi gets, in a given month, as a function of the number of dollars she makes. 𝑏 𝑚 = !

!"

c. Write a composite function 𝑏(𝑚(𝑘)) that gives the number of bills Rashmi gets, as a function of the number of people she kills.

𝑏 𝑚 𝑘 =100𝑘 + 1,000

20 = 5𝑘 + 50

d. If Rashmi kills 5 men in a month, how many $20 bills does she earn? First, translate this question into function notation - then solve it for a number. 𝑏 𝑚 5 = 5 25 + 50 = 75

e. If Rashmi earns 100 $20 bills in a month, how many men did she kill? First, translate this question into function notation - then solve it for a number. 𝑏 𝑚 𝑘 = 100, then 5𝑘 + 50 = 100, so 5𝑘 = 50; finally, 𝑘 = 10 men

Conclusion:

We calculate the composite function 𝑏(𝑚(𝑘)) by substituting 𝑚 by 𝑚(𝑘) in the function 𝑏(𝑚), then we simplify the result to get the shortcut. The shortcut is the result of the composite function that replaces the entire chain of dependent functions.

Lycée Jean-Piaget - Neuchâtel

Mathematics 2MPES Chapter 2

Compositefunctions

Exercise4Make up a problem like exercises #2 and #3. Be sure to take all the right steps: define the scenario, define your variables clearly, and then show the functions that relate the variables. This is just like the problems we did last week, except that you have to use three variables, related by a composite function. Your solution, here.

Exercise5𝑓(𝑥) = 𝑥 + 2.𝑔 𝑥 = 𝑥! + 𝑥

a. 𝑓 7 = 7+ 2 = 9 = 3

b. 𝑔 7 = 7! + 7 = 56

c. 𝑓 𝑔 𝑥 = 𝑓 𝑥! + 𝑥 = 𝑥! + 𝑥 + 2

d. 𝑓 𝑓 𝑥 = 𝑓 𝑥 + 2 = 𝑥 + 2+ 2

e. 𝑔 𝑓 𝑥 = 𝑔 𝑥 + 2 = 𝑥 + 2!+ 𝑥 + 2 = 𝑥 + 2+ 𝑥 + 2

f. 𝑔 𝑔 𝑥 = 𝑔 𝑥! + 𝑥 = 𝑥! + 𝑥 ! + 𝑥! + 𝑥 = 𝑥! + 2𝑥! + 𝑥! + 𝑥! + 2 = 𝑥! + 2𝑥! + 2𝑥! + 2

g. 𝑓 𝑔 3 = 𝑓 12 = 12+ 2 = 14

Exercise6ℎ 𝑥 = 𝑥 − 5. ℎ(𝑖(𝑥)) = 𝑥. Can you find what function 𝑖(𝑥) is, to make this happen?

𝑖 𝑥 = 𝑥 + 5, so ℎ 𝑖 𝑥 = ℎ 𝑥 + 5 = 𝑥 + 5 − 5 = 𝑥