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OSU MTH 111 - Act-#13 Composition-Key

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Math%111% % % % Composition)of)Functions% % % Name:__________________%GrpAct%–%#13) ) ))))))) ) )))))Sect% 12/4/16%%%1. In%the%U n ite d %S ta te s,%t emperatu re %is%c o mmonly%measure d %in %d eg r ee s%F a h re n h e it.%However%in%most%sciences%temperature%is%meas u red %in %C e lsiu s %o r%K elv in .%T he %function%C(F) =59(F − 32)%accepts%an%input%of%temperature%in%degrees%Fahrenheit%and%outputs%the%corresponding%temperature%in%degrees%Celsius.%The%fun ctio n %K(C) = C + 273accepts%a%temperature%measured%in%Celsius%and%outputs%the%corresponding%temperature%measured%in%Kelvin.%%a. Use%the%formulas%above%to%complete%the%following%temperature%conversion%tables.%)Fahrenheit)Celsius))Celsius)Kelvin)%–%13%%–%25%%–%25%248%5%–%15%%%–%10%263%32%0%%0%273%41%5%%15%288%59%15%%25%298%77%25%%40%313%%%%b. Fill%out%the%table%below%converting%temperatures%directly%from%Fahrenheit%to%Kelvin.%%%%%%%%%%% %Prerequisite)Skills)Key)Terms%Learning)Objectives%• Evaluating%functions%• Determining%Domain/Range%• Graphing%a%function%point-wise%• Composition%of%functions%• Inside%fun c tio n %• Outside%function% %• Use%composition%of%functions%to%model%multistep%processes.%• Compose%functions%defined%numerically,%graphically%and%symbolically.%• Interpret%th e % meaning%of%the%output%of%composed%functions%• Determine%the%domain%of%a%composition%of%functions%Fahrenheit)Kelvin)-13%248%32%273%41%278%77%298%104%313%122%323%%c. Give%a%formula%for%a%function%that%will%accept%an%input%of%temperature%in%degrees%Fahrenheit%and%will%output%the%corresponding%temperature%in%Kelvin.%%!!K C(F )( )= C(F )+ 273 =59(F −32)⎛⎝⎜⎞⎠⎟+ 273=59F +22979%%))))d. Use%the%formula %yo u %fo u n d %in %p a rt%c%to%determine(K  C)(50),%and%ex p la i n %what%this%r e p re s e n ts .%In c lude%units .% (K ! C)(50) = K C(50)( )=59(50) +22979=250 + 22979=25479= 283°K%% OR% = K C(50)( )=59(50 − 32) + 273 =59(18) + 273 = 10 + 273 = 283°K%%%%A)Composition)of)Functions%The%diagram%below%illustrates%the%composition%K  C( )(F) = K C(F)( ))) ) ) !!!!!!!!!!!!!original input à fnt à output/input à ftn à final output F↑→ C → C(F)↑→ K → KC(F)( )↑ Temp in Fahrenheit Temp in Celsius Temp in Kelvin ))))))))%![Check!Yo ur!Work].!You!sh ou ld!g et !(K  C)(50) = 283.!!Make!sure!to!fully!explain!what!this!represents.!%%% %The%functi on%you%created%in%part%(c)%is%the%composition%of%the%functions%C%and%K,%and%is%written%as%.%This%is%read%“K!composed%with%C!of%F!”%or%“K!%of%C%of%F.”%In%a%composition,%the%output%of%a%function%becomes%the%input%of%another%function.%%%%%%%%%%%%%%%%%%%%(see%the%diagram%in%the%box%below).%2. The%functions%f%and%g%are%graphed%below.%The%function%f%accepts%as%input%a%UV%index,%and%outputs%the%time%it%takes%for%the%skin%of%an%avera ge %perso n %to%be gin%sho w ing %dam a ge%fro m %sun bu rn .%The%fu nc tion%g%accepts%as%input%the%time%of%day%(measured%as%hours%after%6%am)%and%%outputs%the%UV%index%for%a%typical%spring%day%in%Waikiki,%HI.%%%%%%%a. Find%and%interpret%f(7).%Includ e%un its. %!!f (7) = 12%This%means%that%when%the%UV%index%is%7%it%only%takes%12%minutes%until%skin%damage%begins!%%b. Find%and%interpret%g(10).% %!!g(10)= 4%%This%means%that%10%hours%past%6%AM %(at%4%O’clock)%the%UV%index%is%4%UV%units.% % % % %% % % % % % % %c. If%a%person%starts%sunbathing%at%2%pm%in%Waikiki%how%long%will%it%take%to%begin%show ing%sunb urn%dam age?% %% % % About%7%minutes!% % % % % %%d. Sketch%a%diagram,%like%the%one%in%the%box%for%problem%#1(d),%illustrating%the%compo sition %of%fun ction s%!! f ! g( )(x)%% t↑!→ g → g(t)↑!→ f → gf (t)( )↑"#$Hours since 6am UV index units Time until skin damage (in minutes) %%e. Find%and%interpret%!! f ! g( )(x).%%Include%units.%!! f ! g( )(10)= f g(10)( )= f (4)= 17% %This%means%that%at%10%hrs%past%6%AM%(at%4pm)%the%sun%will%begin%to%damage%your%skin%in%17%minutes.%%f. On%the%axes%at%right,%sketch%a%graph%of%f  g( )(x).%Place %ap p ro p ria te %la b els %(in clu d in g %u n its) %on %e a ch %a xis . %%Use%table%below!%%% % %%%%g. Explain%why%the%shape%of%the%graph%makes%sense%in%terms%of%the%context.%%Because%moving%forward%from%6%am%the%amount%of%time%you%can%spend%in%the%sun%before%damage%gets%shorter%and%shorter%until%6%hrs%past%6%am%(noon)%where%it%is%5%minutes%and%then%it%starts%to%get%longer%and%longer.%x!1"3"5"6"7"9"11"f  g( )(x)"45"17"7"5"7"17"45"3. The%tables%below%define%two%functions%f%and%g.%%x!–)3)–1)5)3))x!–)4)2)0)–)3)f(x)%2)0)4)–1))g(x)!3)5))–)6)–1)%a. Compute%each%of%the%following,%or%explain%why%it%is%undefined .%% g( f (−3))%!!= g(2)= 5%% % f g(−3)( )%!!= f (−1)= 0%%% g ! f( )(2)%!!= g f (2)( )= g(??)%%% g f (5)( )%!!= g(4)= ??%%Undefined!% % % % Undefined!%%% g f (−1)( )!!= g(0)= −6%% f ! g( )(−4)%!!= f g(−4)( )= f (3) = −1%%%%%(NOTE%for%p arts %(b)%an d%(c):%Th e%fu nctio n s%f%and%g!are%defined%by%their%table!)%b. What%is%the%domain%and%range%of%f?%%!!Df= −3,−1,5,3{ }Rf= 2,0,4,−1{ }%%%%c. What%is%the%domain%and%range%of%g?%%%!!Dg= −4,2,0,−3{ }Rg= 3,5,−6,−1{ }%%%d. What%is%the%domain%ofg  f( )(x)?%Need%help?%Sketch%a%diagram%to%illustrate%this%composition.%%!! Dg! f= −3,−1{ }Rg! f= 5 ,−6{ }% %Thinking)about)the)domain)of)a)composition)of)functions))This%d iagram%illustrates%a%composition%of%functions% %%)%%If%x%is%not%in%the%domain%of%f,%then%f(x)%does%not%exist,%and%g(f(x))%does%not%exist.%So%if%x%is%not%in%the%domain%of%f%it%is%also%not%in%the%domain%of% .%Similar ly,%if%f(x)%is%not%in%the%domain%of%g%the n%g( f(x))%does%not%exist%and%x%is%not%in%the%domain%of% .%%%4. Let%21()=fxx%and%() 2 9=−gx x.%%a. What%is%the%domain%of%f?%!−∞,0( )∪ 0,∞( )%%%%b. What%is%the%range%of%f?%%%!0,∞( )%%%%c. What%is%the%domain%of%g?%!9,∞⎡⎣)%%%%d. What%is%the%range%of%g?%%%!0,∞( )%%%%%e. Give%a%formula%for(())fgx.%%!!f g(x)( )=14x − 36%%%%f. What%is%the%domain%of%f  g?%%!! Df !g= 9,∞( )%%%%g. Explain%how%you%found%the%domain%of%f  gin%part%(f).%%First%I%found%the%domain%of%the%inside%function,%g,%then%I%looked%to%see%if%all%of%the%outputs%of%g%could%bre%used%as%inputs%fo r%f.%%T h e %o u tp u t%o f%0 %fro


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