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### Transcript of Lesson 5-1 Operations with Functions

9
Function Composition and Operations New from Old Lesson 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 and Stephan 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's earnings in dollars for working h hours and a function 5(h) to represent Stephan's earnings in dollars for working h hours. h =- hours
i(n):=:. 10h -t(h) =:Tovis car S(h)= gh SCh) ==:stepmn 2. 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 earnIngS for i:he 4- houv jOb 1<;: \$7~.
4. a. Add the functions t(h) and 5(h) to find (t + 5)(h). Then simplify the function rule.(i+s)h = -/:.{h) +- S(h) exp\a In 5.
1(/0;" e)h-I Ibh +- 8h = }8h b. 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?
MATH TIP I- Addition, subtraction, multiplication, and division are operations on real numbers. You can also perform these operations with functions.
The notation (f + g)(x) represents the sum of the functions f(x) and g(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 algebraic expression, remember to subtract each term of the expression. For example, subtract 6x - 2 from lOx as follows. lOx - (6x + 2) = lOx - 6x - 2
=4x-2
The notation (f - g)(x) represents the difference of the functions f(x) and g(x). In other words, (f - g)(x) = f(x) - g(x).
Lesson 5-1 Operations with Functions
7. How much would Jim spend on Tori and Stephans earnings for a job that 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, charges customers 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 for 0\ 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 frees Jlms 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 -h function 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\( e s.a~ \NYlen *=h e1use. Z -J\V'f)S CDrnpa'()'1·
12. Look for and make use of structure. Givenf(x) = 3x + 2, g(x) = 2x - I, and hex) = i2 - 2x + 8, find each function and
nil'? simplify the function rule. --l-~~.....pl~~~~~~~' 0 a. (f+g)(x)
c. (h +j)(x)
e. (g - j)(x)
b. (g+ h)(x)
-t(j():: 3x+2 h he. ::: t\v'W ao '1DU )Q-)D#'.
qCx)-= @x-l \ ine 'K(x) =- ~'2..- 2)(+8 qyodyahc -Wh'l?
o. (~+%)(x}=-f(X)+0(X')- ::::'3x+d. + ( 2)(- \) = ta+ d)(-'-P+ ::::5x+ \ --