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Function Composition and OperationsNew from OldLesson 5-1 Operations with Functions

Learning Targets: ~• Combine functions using arithmetic operations~• Build functions that model real-world scenarios.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge,Discussion Groups, Debriefing, Close Reading, Think-Pair-Share,Summarizing, Paraphrasing, Quickwrite

Jim Green has a lawn service called Green's Grass Guaranteed. Tori andStephan are two of his employees. Tori earns $10 per hour, and Stephan earns$8 per hour. Jim sends Tori and Stephan on a job that takes them 4 hours.

1. Model with mathematics. Write a function t(h) to represent Tori'searnings in dollars for working h hours and a function 5(h) to representStephan's earnings in dollars for working h hours. h =- hours

i(n):=:. 10h -t(h) =:Tovis carS(h)= gh SCh) ==:stepmn2. Find t(4) and 5(4) and tell what these values rry!esent in this situation.-l:(Lf) =-/0 -4- = $'10 wnlS mrnJrgs.fOra-

tf noUY Job are $l}O.SCLf) - 8 4- ¢32 9+ep~h 's eClYnJY~S for_. - a If houv.Job are ~~,

3. Find t(4) + 5(4) and tell what it represents in thjs situation.t (4-)+ '~(4) = .4-0 +32 = 7a

-che Sum of TorI ahd stePhan)S earnIngSfor i:he 4- houv jOb 1<;: $7~.

You can add two functions by adding their function rules.

4. a. Add the functions t(h) and 5(h) to find (t + 5)(h). Then simplify thefunction rule.(i+s)h = -/:.{h) +- S(h) exp\a In 5.

1(/0;" e)h-I Ibh +- 8h = }8hb. What does the function (t + 5)(h) represent in this situation?

The amount: In dollars '+h4t -Itrn muS+-8pett:J on a Job -thcrr +l.\kes h hours.

5. Find (t + 5)(4). How does the answer compare to t(4) + 5~)?

(-t+ 5)4-= t(4-) +SeLf) == 40 of 3'J -= 7~101- if )(Lf) =- 18 (4-) :: ry~

/heY ~re -the Same.6. How much will Jim spend on Tori and Stephan's earnings for the

4-hour job?

S7~

MATH TIP I-Addition, subtraction, multiplication,and division are operations on realnumbers. You can also performthese operations with functions.

The notation (f + g)(x) representsthe sum of the functions f(x) andg(x).ln other words, (f + g)(x) =f(x) + g(x).

Activity 5 • Function Composition and Operations 73

ACTIVITY 5 _,%t) (I: ().s,Y-"":">G\V"';"

When subtracting an algebraicexpression, remember to subtracteach term of the expression. Forexample, subtract 6x - 2 fromlOx as follows.lOx - (6x + 2) = lOx - 6x - 2

=4x-2

The notation (f - g)(x) representsthe difference of the functionsf(x) and g(x). In other words,(f - g)(x) = f(x) - g(x).

Lesson 5-1Operations with Functions

7. How much would Jim spend on Tori and Stephans earnings for a jobthat takes 6 hours? Explain how you determined your answer.

(t+S)h ~ -C{h) -I-S(h) -(10+8)6 = /g(b) =: $/OB

For a basic tree-trimming job, Jim charges customers a fixed $25 fee plus$150 per tree. One of Tim's..competitors, Vista Lawn & Garden, chargescustomers a fixed fee of$75 plus $175 per tree for the same service.

8. Write a function j(t) to represent the total charge in dollars for0\ trimming t trees by Jim's company and a function v(t) to represent the

-l--HI-tl......H.~Y"l~t.f:.JP..jU,t1Q1~f-II.<~'J total charge in dollars for trimming t trees by Vista. t-:f:I:of freesJlms Com~'(Iy j (t..) = a5 + 150(t) -VlSfzt ~ (-l:) = 75"-+ 175(10

" 9. a. Subtract j(t) from v(t) to find (v - j )(t). Then simplify the -hfunction rule. ,\' ( I"\ SL1P.tr:~en I Ire.C\V:~-j.)l-F) =- V (+) - J T) /. ~uo Y\ iy ~= i5+I"lS(-t') - (05+ 150(-\:'»)

s: -, S"-l- \75 (+) - 2 5 - 15at+")b. What does the f~cti~9v ~~~ r~t)sent in this situation?1he arf'lCUn+ dr mone,! ~ CUt;;.-rorneV" \(\j\\ \

So.\Je b'l US1Yl5 3\VY1S C1Jmpan'l \Y1S-tetlQ, or'\f Istzl.. - ~

-+-+-+-I---+--+-+--+-+-+-f:--i-':-'""")X:::-;Crdr A = 100 \00 -so = ao 'Iou save 7.0 Be ax.i.+ B= qo80 b'l buy 1r)9 Coa+ .• Find (v - j)(5). What does this value represent in this situation! '\ $q5t)

(V-j )(5) ::.,\Ie s) -J(5) .".(5)= 15+ \15(5) == ~sO-\-dSlls) '= =so- 115 - j (S) z: as+ \SO(s)~11S*

~'),,-' $ - s\"Omer \N \\\ save $ \15 ~5O~d5J~'lS= \l5J?f~e'l chd)Se 1{mS~t'Y2P~Y1+v1Ies~sq1. ~ow much will a customer save by choosing Jim's companjq;tnm • ~

1.7"t'~~""""'I~r""""'T"""""""""""""""""''''''''''''''''''''' 8 trees rather than choosing Vista? Explain how you determined your ~

~-+I-:-+---f--":+--J-=-F-L-+--t.--t--I (V_j~'i)r-\?O+a5t 'X~) $. i= 50+ a5(8) = a50 . 8

-+-+--IL.A.fL-4-J-~~.J1l4r4--+--l\he'l 'N \\\ Sa\( e s.a~ \NYlen *=h e1use. Z-J\V'f)S CDrnpa'()'1·

12. Look for and make use of structure. Givenf(x) = 3x + 2,g(x) = 2x - I, and hex) = i2 - 2x + 8, find each function and

nil'? simplify the function rule.--l-~~.....pl~~~~~~~' 0 a. (f+g)(x)

c. (h +j)(x)

e. (g - j)(x)

b. (g+ h)(x)

d. (j- g) (x)

f. (h - g)(x)

74 Springfsoard'" Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions

-t(j():: 3x+2 h he. ::: t\v'W ao '1DU )Q-)D#'.

qCx)-= @x-l \ ine'K(x) =- ~'2..- 2)(+8 qyodyahc -Wh'l?

o. (~+%)(x}=-f(X)+0(X')-::::'3x+d. + ( 2)(- \)= ta+ d)(-'-P+ ::::5x+ \ --

{ lh-~)()()"" h(x)-0(x) - - - --~- -- .~-~-)(~ - d x+8 .: (2x-\) ---- ---_::._x'2- a)< t 8- d)( ~ I _---C)(~-4)(+y -- -----