Jim has been asked to make a bid for installing the shrubs around a newoffice building. In the bid, he needs to include the number of shrubs he caninstall in an 8-hour day, the cost per shrub including installation, and thetotal cost of his services for an 8-hour day. \f\l \1rtff----t+-i-.J.r---..~~tl13. a. Write a function n(h) to represent the number of shrubs Jim car/ .

install in an 8-hour day when it takes him h hours to install,..qrfe I'

shru_(b.:'\ ~ h n ('t!) When considering restrictions onIn J z: 'n \ 8 the domain of a real-world

~ 2 I\ function, consider both values of. '1 . the domain for which the functionL\- 'l.. would be undefined and values of

• . . 7 . 0 \ the domain that would not makeb. What are the restrictions on the domam of n(h). Explain. 0 sense in the situation.n? houl{s OiVld YlDUI('S mUS+ be ~ .L ~~~~:p:::::::::r::::;::p::=+=+=l-Jfl>osrnve (couQd c..o vYl(I bock- ~ ~\\he \lo.\ue at h cannot oc 0) O( \NouiJ'oe UVlO rea .

14. Jim will charge $16 or each shrub. He will also charge $65rer hour forinstallation s rite a unction c(h to re resent the amount Jimwill charg or a shrub at ta e .ours to inst,a. h \nptd-

nth \6 +",b - 1_\G:. - '\sth.e.\na~oen1\jC\'(\aD~1 I I I I I I I I I

~~~ 'NhOtt \8 -\-he ~~-ed.

Lesson 5-1Operations with Functions

(A\\ \mpt-\N6(OS,

unoer\\n-e.~

r>

.,;<II~<IIVI~VI

:E.~

The total cost of Jim's services for an 8-hour day is equal to the number ofshrubs he can install times the charge for each 'shrub.

15. a. Find the total cost of Iims services using the functions n(h) and c(h)to find (n • c)(h). Then simplify the function rule.

(n p 0(h) ~ DCn)' c(n't '\~~ .. (bS"ntlb)

~ . 2)('n) _ h 12~ + 520b. Attend to precisio-;. What are ~e restrictions on the domaim of

(n • c)(h)? \(-) ~ 0h>O --;-n \\lUst- be PJs\n~ic).o,~

0'00. 'N!\r\ ~

continued

«'E'"oco<II

""2-0u,'"...oN

<9

r=>;

The notation (f • g)(x) representsthe product of the functions f(x) .and g(x). In other words,(f. g)(x) = f(x) • g(x).

Activity 5 • Function Composition and Operations 75

~-~""::r " .~-~

[ACTIVITY 5 j~" Lesson 5-1 -

continued ""l' ,- Operations with Functions

-- - --II' 'ft,' .11:.1.' .r--.•.

16. Reason quantitatively. Jim estimates that it will take 0.5 hour to- r---- \ install each shrub. Use the functions n(h), c(h), and (n • c)(h) to- r---- 1«' I II!) BUSINESS determine the following values for Jim's bid, and explain how you- - determined your answers. , -When a company makes a bid on a. the number of shrubs Jim can install in an 8-hour day- - a job/ the company states the price h(h) = -t ~fa- := 110 £hvll\'c6-- at which it is willing to do the job.-- The company must make its bid-- high enough to cover all of its- - expenses. Ifit bids too high,- - however/ the job may be offered to b. the cost per shrub, including instal~tion

one of its competitors. ~(h)=- \~ +65 (Vaf 4-8,50'-It IYI p.,d- \+ ~'les 1/2..'<Iv pur 5hYL\ bhoU,(S

- c. the total,•.cost of Jim's services for an 8- ur day

(0-Ci.h)=B-.(\b+bSh ='() :\ 1'09 -\-52D = T1.b

\j('()) 0 Ceh).:. (n,CJ.'5J= :ۥ J' -

17. Explain how you could check your answer to Item 16c.t'J\~ \:; ~e:;r of ShyubW,m caY)

\'()<;~ \\ \'(1 ~n g-h'( dC\"I by c.o~ perShI(U'o) \ndud\Y/0 \h~I\Gtilon, lb~ltg/3);:71b ~-

Jim offers two lawn improvement services, as described in the table,

Lawn Improvement Services

Service Hourly Material Cost forCharge ($) Average Yard ($)

Compost 40 140 -ci.,1:.,

Fertilizer 30 30 "'~- ~- s:

18. a. Write a function c(h) to represent the total charge for applying.~<C

compost to a lawn, where h is the number of hours the job takes. -ci:;;c en) =: ~Dh + \\.\-0 0

<Xl

'"""~-0u'"""aN

b. Write a function j(h) to represent the total charge for applying 9

fertilizer to a lawn, where h is the number of hours the job takes.

feh) ~ 3dn+ 30

,r-'

76 Sprlngfloard" Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions

I

Lesson 5-1Operations with Functions

I·~·

r:

._,.,.-- ~---.---,t '

.~""•.'II'-.'. ATH ,

----~----.--~~~

~~-.pf·~ - -~. -:-'~.:v ,r:" ~ r. ;:. ~:~.- .• -l ACTIVITY 5 J

*' continu;d

The notation (f -;-g)(x), g(x) 7= 0represents the quotient of thefunctions f(x) and g(x) given thatg(x) 7= O.ln other words, (f -;-g)(x) = 1-1--

f(x) -;- g(x), g(x) 7= o.

b. ~h6~~n \~t~~(~~nt .in-this situation? ~

'\ J..kY'J.)r0.'oeJf d\" IS\ 0 V) \ ()"~ ck ~ Ii .'('ern D df 4he LOS;- In d oj laYS To Otppl'll;erGO rnpost- -ro ~ wst \Y) d o\\aV"S ~f +erh \I'ZC('.

20. Find (c +j)(4). What does this value represent in this situat~

C(41=4-0(4)+\4-0 == d ~~ 9~~r5 ~e-f (4) :3a( >.\)+ "30 COSt? ~ ~Stf 1"tW-l~L.II-+-----t----I--l-+----+---l---l-

\s ~tw'nCS ~21. Look for and make use of structure. Givenf(x) = 2x,

g(x) = x + 3, and h(x) = 2x + 6, find each function and simplify thefunction rule. Note any values that must be excluded from the domain.a. (j. g)(x) -=::- ( d.",-V+3) = '2 y....?' +6Y--.

~ 1),''R

'--I---

I--

.---

You may be able to simplify thefunction rules in Items 21c, d,and e by factoring the expression'snumerator and denominator anddividing out common factors.-ci

~•.'"~1).s:

""-c

<t'E'"ocog:,~(5u'"....•oN

19

+----l-e

7\~O) \R.

~

~ Activity 5 • Function Composition and Operations 77

,/8@ operahOns DO _fea}nuVJierS \'()\fd\le O'O\,/-

numberS. ~orS on -funchoos IO'l)\'IC ----fuY\tt\on (u \-eS· -,me VroC-ess of crl:hrvn) subtfamol'\) rru \tt~\lCOiTiJyLand d \\}\S\On ore essen-ra \l'j tye :?an'\t! -fur dlVISlDn of fCtA\ rum~) ~ d\ \sor_- - - --c1Al{l'f\CJr 'o-e 0) OtnJ for _dt.\JlSlon oF -fundlonS)~ .fuVlLtlon '\U\-e ~ \$ --\he diVisor Cl.\hrot -

_ be e?oLtAl 1'D o.

'a3 I£:p.;:o_ ~x+L 'I..:j:d.~) 1R~9 -.3-)(--aMa\l1DYl D~ func.;hcrG \CS CJ)~mu~fl-t

fl~)=4~+ -- ----L~") ~ -dxt"5 - -

_( +~).,. (9+iCXX) ~ 'dx +-'l --- -- ---

s nL)()"= ~><+5 d-C)()--( h-j'X~)-:.x-'} -- - -

~~JOO=-X-d ---=- --- -=--- -i)--X-t-a::= j(x) -

_ 4~tS- X +~ :::3'kr-t ""j(><-) . ==

lab C-F+9Xi>:: 5X+ i- 3x- = gx-3 .. -~ (.r-~.)(~}",Sx+\ - (3)<.-4-) = 5)(+-\-3)(-\-4 ~ -

= dX+S -

- - - ......:.-

Cle_\f"~XX);; (5x+l; 3)(__+)= - ---. \'5~~+3~-~b~--+= - - -

\S~d-\lx- 4-- --

x\: 40 )1R ·-a:l (-FXy-) = 5x+l~ 3~-4-~o (3-fXJ<.)-3x-4--(5x+i) ._ ". pc'Ycn\::Y'CS\S

-= 3~-4--5)<:-\_ _ ~ __ = -2x-S -

3\ a(+)=-80Q+84-tf(t.} -::;~b4-t

CO lCHf)( 1-') = 800 +24t -t~4t =800+ ~4 8-\: .

The furdloh( a+f)t rq>res-enis -\:he (Dst- ---cJ;. 'f"nG\V--I~ -We-ad aYld ruvwll~ It 00 ootn. .S~-nons t t\mes, - - -.

b (d+f')lt)::: (dtfXr~)~gj)-t 34€{\~)=800 + tt\76 ::.'0/1~ .

-:t-t.\I'j\\\ WST $4'116 -tD ~ ~ add-a run -It on to%) ~-hChs la.-hYhe$.