00020002010015Inverse_Functions.ACT0001020011Inv_Functions.EAC010000002d63É řúy=2^(x+0.2)+0.25ü÷Î÷ĘČ ln(íëx-0.25)/ln(2)-0.2 ü ř úy= log(2,x-0.25)-0.2 †ŠdI)Wr†`0†Š`Œu† )™™™™™Š †!ŽDvŠIˆK†ˆC A† ŽkLŽ'ˆxC˜ ˆ_sx˜yŽ ˆ ˆq †šČŔ  Š@†ČĐ6–°†Ďgf†ÚY™”  ˆ°ŽFŠ †űĘȐB‡ 2^(íëx+0.2)ˆ5††qŠâˆ/ˆ ˆy‡5ˆ‹:  Inv_Functions.EAC††erse–ACT†.†1†;ˆ>†BŠ†U `E Ž'C[ˆ4 Explore Žb ’{Ž!\Œ) Authoríˆ@†D펁„†ˆ˝ Wei-Chi Yang Œ Radford UniˆĆity Ž/ e-mail: wy†9@rŒ2.eduŽE!URL: http://www.–#/Š5ŽÓŽ Obje‡Gves:[Ž O(1) We will eŽűthe ia f‘z †ł some trigon† † c–!and Œö@exponential functions.[E(2) We will also see how shiftingsŽffect the inverse¤MŽU Part I. 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What is‹B f‡A‡7+î)?4Q2.)#*What is the inverse for -f(x-î)+1?Rˆdefine †)=siníü(x)RŽ done[Œ,\Œ4Explore Q1 here=>Ј!†%Ęˆ]NH%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ ŕxy†^‰‡(ěˆ(Hˆ2‰hŠ‡(ˆ´†ź(ä<†7ŒČäŽP T‰+ †ä† † ‡l)`’’l’’x’’„’’’’†ô"’¨’’´’’Ŕ’’̒†@ؒ†ä††đ††ü† ‘† !‘,‰Œ"‘@ˆ2† #‘T,†$‘h8† %‘|D† &‘P† !'‘¤\¸†ä(†h†  †)†it†m*†€†+†Œ†ˆW˜†-†¤†.†°†0†ź†1†Ȇ2†Ԇ3†ŕ†4†ě†5†ř†€E‘† F†Ę† H‘,† I†É‰J†ÉJ‘Tˆ‚† K†Ę@†L‘|ˆ †M†ËX†N†Ëd‡-O‹¸†Ëp††Ć†Ú P††|†wQ†{ˆ†R†”†S† †T†Ź†]‡p ¸†} ˆ^ŽĐ_†Ŕ†`†ĆaŽŕb†̒ˆŞŠßВŒ•†܆͆č†Βiô’ˆFŠ‡DŽב@ ’ؑT’ّh$’ڑ|0’ۑ<‘¤ Financial‡žFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ bŒˆxRü‘ŽŽžž ţ@ţ€ţŔÓ††ˆŽŢ0ţ`ţ ţŕ˙ ˙`Ă Šœ6 ˙˙@˙xŽ ž!ˈŹ%†† a‡. MatDatab‡:.EAC‹A‹F† ” ”ÁžŠż  —€† ††CĐ<ˆ^†¸ ˆˆ † –ňŒň ö<ö<ţxţđ˙,˙h§¤]ˆ““ ^䆐ˆ‰ˆ ˆ  ††††\ŒBŒH’ ’,† †0†!@†@PSheet1|đ‘|ŞŒ`Š2ž3ž 4†5ž †Çä8ä7Ę|č–|œš!˜|†x “i™” ™‰G‰K† 7“-ˆ‰ˆ`E ‰eŽŔ‰`†$Ťd‹ť†Œ” †† Ž"†fn’$†(œ†AŠ  †j 3xf(x+ä)”/ˆkŽxŒ†0 ŕL(x)-ä|(† )/2ŕxŕä/2ˆƒŽ/`– PŘĆ$´$“†S‚@Œ ˆ!†–„“ ˜€–`’ ˜0™Š(1…0qy`‰@#Y‡uYƒŠH–‘b†† Š – ˜Ć$ĘHDŽ `E…Ua!Š ’#’,¤˜°´âIâzâŤÚܘ؏đ`‘ą ˜ ’e5‰y`” – œ †ˆ$†Yˆ †[Š\ŒExplore Q2 here=>Ј!†%ĘŒ5\NŒFinaForm$N†Graph2D†, 3Š@ ˆLISTSYSˆL4ˆ< Modify €ˆPˆ<STATCALC ¤ˆdˆŒ< Ź\ˆx S:equenceˆŒ,ˆxSheetˆO4|’ŠŒ°’ olveEqˆ´†€`wr0ˆ´(Upˆ<’tupFLG1H(Š<†Lis†{pD‰ˆPic†´ÜViewWindŘˆŒ_osve†v‡!ôxy†^‰‰ †i†y2’ 8Š‡’X‰|ŠˆúŒl<äŽP Œ‰( †ä† † ‡i˜’’¤’’†Ů@’ź’’Ȓ’Ԓ’ŕ’’ě’’ř’´’Č’Üˆ Œđ(†‘4† ‘@†  ‘,L† !‘@X† "‘Td† #‘hp† $‘||† %‘ˆ† &‘¤”¸†ä'† †  †ˆlŹ†ƒ)†¸†*†Ć+†І,†܆-†č†.†ô†ł0 ˆŠ†ł1´ †2Č† 3Ü$† 4đ†%Z†Ř5†Ď<†E†ĎH† F†Ď T’ˆ‹C`†I†Ďl† J†Ďx† K†Ď„† L†Ď† M†Ďœ‡3N‹¸†Ď¨††Ę†Ţ O††´†ŁP†§Ŕ†Q†̆R†؆S†ä††yđ†]‡~ü†‘ ˆ^Œ†¤Ž_ †Â`†Ý† a†ÝˆćbÜ† ”‡Ť’Œ•‘ ’͑,† Α,8† Б@D† בT‰JŒ‰"‹k\†ّ|h† ڑt† Xۑ¤€‡˝ "† FinancialFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ "bŒˆxRü‘ŽŽžž ţ@ţ€ţŔŤü‘Rœ ˙˙@˙x‡¸ŽüŽŒ< WóˆŹ%†† a‡F MatDatab‡R.EAC‹Y‹^† ” ”ÁžŠż —˜† †¨<ŠÝ^†]ˆíˆŒ’ ‘ˆ Œˆ(ˆ",ö<ţxţ´ţđ˙,˙h—¤†œ‡X‡[–eŒ’^Œ ˆ0Rü‘ˆ †††† †ŽTŽ[Ž ’,† †0†!@†@PSheet1|đ‘|ŞŒ`Š2ž3ž 4†5ž ä8ä7Ę|č†]–|œš"8˜| “i™” ™‡€‰ƒ‡‡“-‘—`E ‰eŽŔ‰`†$¸ˆ6†ŻŒ˜ ¤†˝†† ˛6’$†(œ†AŠ  †~ 3x-f(x-ä)+1–3%Œ † ŕL(-†"1))+ä|(ˆ+ /2+1ŕxŕäˆ –8ˆ§ŽXxŽ†-ŕM–m†łŽ#`– P‘Ć$´$“c†S‚@Œ ˆ!†–„“B ˜€–`’ ŒÄ‡s1’ ˆ(1…0qy`ˆ#Y‡uYƒ†* ˜–0– šHŽĆ$ĘHiEVRŠ´7A(%Q˘ŔšÍŽ°´âIâz⍲܆ŒŽ ’ Ž`: ˜’e5‰yˆ”†– œ˜`†0Zˆ4†8[Œˆ Remark:Œ($ Assume a function f has an inverse,[†=ˆ„! a) If f is shifted to the right Œd a unit(s)Ž8-, its corresponding_žLupŽEaŽNŽL”‡b˛‡lef˛†/Ԇdown˛ˆRW2c) If f is reflected along x-axis, the correspondi†inverse¨5yŠ5 .[[Œ TPart II. Ex†Fential Functions: W†dan check†ratŠwf! f1 and g1 below are Œd8Ž of each o†ą r, which w˜X algebraically’¤RˆŠŒ° a:=2RŽ† †ŽĹdefine f1(x)=aŒäx–1done˜4 Ž4gŠ43ŒoaŞ;< simplify(f1(Š?))˜›†w† g1(f1(x))RŠx[†  Exploration:†ˆ2KStep 1. Drag the followin†wo geometry links with y=x int–strip:]†ƒˆ Ž]†*[ŒiUŠi 2. Move only†%e graph at a time and figure out whatŠ–o†›rŠ2 should be so Œâ#†ˆ…ay†˜verseŽŞeachŒ;.\ 7Šč–zœp‰ correspond‰ŽQΉaĎTStep 3. After you are su†ofˆ r choicesˆ?expressions, define them as f2 and g2 Nrespectively belowŠ!verify algebraical†ifˆHyŠ}inversŠ€each o†br.RˆTŒ[Ž{ f2(x)=2Œrx+0.2Œ~ˆ 5RŒ4done˜E ŽEgŠEŒDx-†D5ŒĂŽMˆŞLŒësimpl†Ö(f2(ŠP)–zxŞ+g2ˆ.ś+[†Ť‰L Exerci‡1.]  If g(x)=î-xŒ ", What is the inverse for†3-x)+2?Rˆ1Œ8defineO”NRŒ$doneŠ5Œl\Œt look hereЈ†ĘˆN8%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ ŕxy†^‰‡(ěˆ(Hˆ2’‡†y$‰|ˆ(ä<‡hŒČäŽP ‰: †ä† † ‡iTP’’\’’h’’t’’€’’Œ’’˜’’¤’’°’’ź’’Ȓ’Ԇ†ŕ††ě† †ř†!‘,† "‘@† #‘Tˆ–Œ$‘h(†%‘|4† &‘@† !'‘¤L¸†ä(†X†  †)†_d†c*†p†+†|†,†ˆ†-†”†.† †0†Ź†1†¸†2†Ć3†Іˆ}܆5†č†E†ô†ˆF‘‰!†­H†É †I†É† J‘T$† K‘hˆć†%L†É<†M‘ˆn†N†ĘT‡&O‹¸†Ę`††Ĺ†Ů P††l†nQ†rx†R†„†S††ˆ;œ†]‡i ¨†} ˆ^ŽĎ_† °’ˆŞŠŁ´† aŽŕb†źŔ’Œ•†̆ ͆؆Ά䇻Đ’đ†׆ ü’ˆF‹W†ّh† ڑ| † 5ۑ,‘¤ Financia‰ĽFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ bŒˆxRü‘ŽŽžž ţ@ţ€ţŔÓ††ˆŽŢ0ţ`ţ ţŕ˙ ˙`Ă Šœ6 ˙˙@˙xŽ ž!ˈŹ%†† a‡. MatDatab‡:.EAC‹A‹F† ” ”ÁžŠż  —€† ††CĐ<ˆ^†¸ ˆˆ † –ňŒň ö<ö<ţxţđ˙,˙h§¤]ˆ““ ^䆐ˆ‰ˆ ˆ  ††††\ ŒBŒH’ ’,† †0†!@†@PSheet1|đ‘|ŞŒ`Š2ž3ž 4†5ž †Çä8ä7Ę|č–|œš!˜|†x “i™” ™‰G‰K† 7“-ˆ‰ˆ`E ‰eŽŔ‰`†$Ťd‹ť†Œ” †† Ž"†f[’$†(œ†AŠ  †j  3xä^x+2–/ Œ†ŕ_(x-2)”ˆ‡Ž8xˆsŽ`– PČĆ$´$“†S‚@Œ ˆ!†5–„“ ˜€–`’ ˜0™z(1…0qy`‰0#Y‡uYƒŠH–˜œ˜HŒş – – ˆ_/p `R‚qƒŠ$’#’,¤˜°âIâzâŤÚܘ؏đ`‘Ľ †p ’e5‰y‰Pˆ ˆ ˆOˆŒœ"ˆ$Y[Œˆ Exercise†ˆ" Œ2.Ž If h(x)=-2Œ@-(x+3)Œ -4. 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MatDatab‡:.EAC‹A‹F† ” ”ÁžŠż  —€† ††CĐ<ˆ^†¸ ˆˆ † –ňŒň ö<ö<ţxţđ˙,˙h§¤]ˆ““ ^䆐ˆ‰ˆ ˆ  ††††\ŒBŒH’ ’,† †0†!@†@PSheet1|đ‘|ŞŒ`Š2ž3ž 4†5ž †Çä8ä7Ę|č–|œš!˜|†x “i™” ™‰G‰K† 7“-ˆ‰ˆ`E ‰eŽŔ‰`†$Ťd‹ť†Œ” †† Ž"†ff’$†(œ†AŠ ˆ  3x-2^(-(x+3))-4 Œ ˆ ŕ`(2,ˆ"4†"3”Wˆ“ŽDxˆŽ`– PÔĆ$´$“†d€Œ ˆ!†9–„!`Y ™ †¨” ˜0™†(1…0qy`‰<#Y‡uYƒ‡Ľ ˜–˜œŒÉ%– ˜Ć$ĘH’ ’¤Č%°´ţaţ ţߞجđ`ˆ`˛ ™’ e5‰y`ˆ ††– œŽNˆ$Y‰#[eActa fHxŕM(x)f1Hxa^(x)f2H$x2^(x+0.2)+0.25gH xä^(-x)g1H x ŕ`(a,x)g2H$x ŕ`(2,x-0.25)-0.2010008main.ACT0001020012eActivity Save.EAC010000002e69É řúy=2^(x-0.225)+0.4ü÷Î÷Ę@" ln(íëx-0.4)/ln(2)+0.225 ü ř ú log(2,x-0.4)+0.225 †ŠŠ ˆ `@”Gs#Š Œ3†Ž!Š †ŽDvŠ$ˆK†r†  A† ŽkLŽ'ˆxC˜ u˜x˜yŽ ˆ ˆq †šČŔ" Š@†ČĐ6–°†Ďgf†ÚY™”  ™ŽCŽĽ †űĘČ !ŒB‡ 2^(íëx-0.225)+0.4†ăˆ0ˆ ˆz‡6ˆ‹;  Inv_Functions.EAC††erse–ACT†.†1†;ˆ>†BŠ†U `E Ž'E[ˆ4 Explore Žb ’{Ž!\Œ) Authoríˆ@†D펁„†ˆ˝ Wei-Chi Yang Œ Radford UniˆĆity Ž/ e-mail: wy†9@rŒ2.eduŽE!URL: http://www.–#/Š5ŽÓŽ Obje‡Gves:[Ž O(1) We will eŽűthe ia f‘z †ł some trigon† † c–!and Œö@exponential functions.[E(2) We will also see how shiftingsŽffect the inverse¤MŽU Part I. Trigonometric F}:\Œ}Bas†ˆ&”ЈL†PĘˆŁ NH%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ ŕxy†^‰ ěHˆ2ˆđŠ‡†|0†ä<†7ŒČäŽP T‰) †ä† † ‡j)`’’l’’x’’„’’’’†ň'’¨’’´’’Ŕ’’̒’ؒ’䆆đ††ü† ‘† !‘,‰Œ"‘@ˆ2† #‘T,†$‘h8† %‘|D† &‘P† !'‘¤\¸†ä(†h†  †)†it†m*†€†+†Œ†ˆW˜†-†¤†.†°†0†ź†1†Ȇ2†Ԇ3†ŕ†4†ě†5†ř†€E‘† F†Ę† H‘,† I†É‰J†ÉJ‘Tˆ‚† K†Ę@†L‘|ˆ †M†ËX†N†Ëd‡)O‹¸†Ëp††Ć†Ú P††|†wQ†{ˆ†R†”†S† †T†Ź†]‡l ¸†} ˆ^ŽĐ_†Ŕ†`†ĆaŽŕb†̒ˆŞŠßВŒ•†܆͆膷şôg’ˆFŠ‡DŽב@ ’ؑT’ّh$’ڑ|0’ۑ<‘¤ Financial‡žFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ bŒˆxRü‘ŽŽžž ţ@ţ€ţŔÓ††ˆŽŢ0ţ`ţ ţŕ˙ ˙`Ă Šœ6 ˙˙@˙xŽ ž!ˈŹ%†† a‡. 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What is‹B f‡A‡7+î)?4Q2.)#*What is the inverse for -f(x-î)+1?Rˆdefine †)=siníü(x)RŽ done[Œ,\Œ4Explore Q1 here=>Ј!†%Ęˆ]NH%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ ŕxy†^‰‡(ěˆ(Hˆ2‰hŠ‡(ˆ´†ź(ä<†7ŒČäŽP T‰+ †ä† † ‡l)`’’l’’x’’„’’’’†ô"’¨’’´’’Ŕ’’̒†@ؒ†ä††đ††ü† ‘† !‘,‰Œ"‘@ˆ2† #‘T,†$‘h8† %‘|D† &‘P† !'‘¤\¸†ä(†h†  †)†it†m*†€†+†Œ†ˆW˜†-†¤†.†°†0†ź†1†Ȇ2†Ԇ3†ŕ†4†ě†5†ř†€E‘† F†Ę† H‘,† I†É‰J†ÉJ‘Tˆ‚† K†Ę@†L‘|ˆ †M†ËX†N†Ëd‡-O‹¸†Ëp††Ć†Ú P††|†wQ†{ˆ†R†”†S† †T†Ź†]‡p ¸†} ˆ^ŽĐ_†Ŕ†`†ĆaŽŕb†̒ˆŞŠßВŒ•†܆͆č†Βiô’ˆFŠ‡DŽב@ ’ؑT’ّh$’ڑ|0’ۑ<‘¤ Financial‡žFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ bŒˆxRü‘ŽŽžž ţ@ţ€ţŔÓ††ˆŽŢ0ţ`ţ ţŕ˙ ˙`Ă Šœ6 ˙˙@˙xŽ ž!ˈŹ%†† a‡. MatDatab‡:.EAC‹A‹F† ” ”ÁžŠż  —€† ††CĐ<ˆ^†¸ ˆˆ † –ňŒň ö<ö<ţxţđ˙,˙h§¤]ˆ““ ^䆐ˆ‰ˆ ˆ  ††††\ŒBŒH’ ’,† †0†!@†@PSheet1|đ‘|ŞŒ`Š2ž3ž 4†5ž †Çä8ä7Ę|č–|œš!˜|†x “i™” ™‰G‰K† 7“-ˆ‰ˆ`E ‰eŽŔ‰`†$Ťd‹ť†Œ” †† Ž"†fn’$†(œ†AŠ  †j 3xf(x+ä)”/ˆkŽxŒ†0 ŕL(x)-ä|(† )/2ŕxŕä/2ˆƒŽ/`– PŘĆ$´$“†S‚@Œ ˆ!†–„“ ˜€–`’ ˜0™Š(1…0qy`‰@#Y‡uYƒŠH–‘b†† Š – ˜Ć$ĘHDŽ `E…Ua!Š ’#’,¤˜°´âIâzâŤÚܘ؏đ`‘ą ˜ ’e5‰y`” – œ †ˆ$†Yˆ †[Š\ŒExplore Q2 here=>Ј!†%ĘŒ5\NŒFinaForm$N†Graph2D†, 3Š@ ˆLISTSYSˆL4ˆ< Modify €ˆPˆ<STATCALC ¤ˆdˆŒ< Ź\ˆx S:equenceˆŒ,ˆxSheetˆO4|’ŠŒ°’ olveEqˆ´†€`wr0ˆ´(Upˆ<’tupFLG1H(Š<†Lis†{pD‰ˆPic†´ÜViewWindŘˆŒ_osve†v‡!ôxy†^‰‰ †i†y2’ 8Š‡’X‰|ŠˆúŒl<äŽP Œ‰( †ä† † ‡i˜’’¤’’†Ů@’ź’’Ȓ’Ԓ’ŕ’’ě’’ř’´’Č’Üˆ Œđ(†‘4† ‘@†  ‘,L† !‘@X† "‘Td† #‘hp† $‘||† %‘ˆ† &‘¤”¸†ä'† †  †ˆlŹ†ƒ)†¸†*†Ć+†І,†܆-†č†.†ô†ł0 ˆŠ†ł1´ †2Č† 3Ü$† 4đ†%Z†Ř5†Ď<†E†ĎH† F†Ď T’ˆ‹C`†I†Ďl† J†Ďx† K†Ď„† L†Ď† M†Ďœ‡3N‹¸†Ď¨††Ę†Ţ O††´†ŁP†§Ŕ†Q†̆R†؆S†ä††yđ†]‡~ü†‘ ˆ^Œ†¤Ž_ †Â`†Ý† a†ÝˆćbÜ† ”‡Ť’Œ•‘ ’͑,† Α,8† Б@D† בT‰JŒ‰"‹k\†ّ|h† ڑt† Xۑ¤€‡˝ "† FinancialFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ "bŒˆxRü‘ŽŽžž ţ@ţ€ţŔŤü‘Rœ ˙˙@˙x‡¸ŽüŽŒ< WóˆŹ%†† a‡F MatDatab‡R.EAC‹Y‹^† ” ”ÁžŠż —˜† †¨<ŠÝ^†]ˆíˆŒ’ ‘ˆ Œˆ(ˆ",ö<ţxţ´ţđ˙,˙h—¤†œ‡X‡[–eŒ’^Œ ˆ0Rü‘ˆ †††† †ŽTŽ[Ž ’,† †0†!@†@PSheet1|đ‘|ŞŒ`Š2ž3ž 4†5ž ä8ä7Ę|č†]–|œš"8˜| “i™” ™‡€‰ƒ‡‡“-‘—`E ‰eŽŔ‰`†$¸ˆ6†ŻŒ˜ ¤†˝†† ˛6’$†(œ†AŠ  †~ 3x-f(x-ä)+1–3%Œ † ŕL(-†"1))+ä|(ˆ+ /2+1ŕxŕäˆ –8ˆ§ŽXxŽ†-ŕM–m†łŽ#`– P‘Ć$´$“c†S‚@Œ ˆ!†–„“B ˜€–`’ ŒÄ‡s1’ ˆ(1…0qy`ˆ#Y‡uYƒ†* ˜–0– šHŽĆ$ĘHiEVRŠ´7A(%Q˘ŔšÍŽ°´âIâz⍲܆ŒŽ ’ Ž`: ˜’e5‰yˆ”†– œ˜`†0Zˆ4†8[Œˆ Remark:Œ($ Assume a function f has an inverse,[†=ˆ„! a) If f is shifted to the right Œd a unit(s)Ž8-, its corresponding_žLupŽEaŽNŽL”‡b˛‡lef˛†/Ԇdown˛ˆRW2c) If f is reflected along x-axis, the correspondi†inverse¨5yŠ5 .[[Œ TPart II. Ex†Fential Functions: W†dan check†ratŠwf! f1 and g1 below are Œd8Ž of each o†ą r, which w˜X algebraically’¤RˆŠŒ° a:=2RŽ† †ŽĹdefine f1(x)=aŒäx–1done˜4 Ž4gŠ43ŒoaŞ;< simplify(f1(Š?))˜›†w† g1(f1(x))RŠx[†  Exploration:†ˆ2KStep 1. Drag the followin†wo geometry links with y=x int–strip:]†ƒˆ Ž]†*\Œ…Έ–[Œ„UŠ„ 2. Move only†%e graph at a time and figure out whatŠąo†śrŠ2 should be so Œý#†ˆ ay†łverseŽĹeachŒ;.ŽŁ7‹–zœp‰ correspond‰ ŽQŒÚ†ŢÎˆě† ‹ŠdI)Wr‡`0ˆ‰™`Œu†  )™™™™™†† ˆv† ††r†  A† 'LŽ'ˆ<C˜ u˜x˜yŽ †uˆq†~ČŔ  Š@†Đ6–°†”gf†ŸY™”  ˆ°ŽFŠ†ŔĘȐBˆÎ 2^(íëx+0.2)ˆ5†Ž&’&lnˆ&-†&5)/† 2)ˆ †™‹ ˆWˆ5ˆĄŠgˆŒË[‰0TStep 3. After you are su†ofˆ r choicesˆ&expressions, define them as f2 and g2 Œ[N)respectively below and verify algebraical†if they are inverse of each o† r.Rˆdefine f2(x)=2Œx-0.225Œ% +0.4RŒ5done˜F ŽFgŠFŒEˆE4Œh2ŒK†K†XŞMŒ’simpl†Ř(†”ŠQ)–{xŞ+†yŠÂ˛+[†Źˆó Exerci‡1. If g‰î‘-x! ", What is‰W“R for g(-x)+2?ŠPŽOO-xRdoneRˆ Œ\Œ look hereЈ†Ęˆ3N8%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ ŕxy†^‰‡(ěˆ(Hˆ2’‡†y$‰|ˆ(ä<‡hŒČäŽP ‰: †ä† † ‡iTP’’\’’h’’t’’€’’Œ’’˜’’¤’’°’’ź’’Ȓ’Ԇ†ŕ††ě† †ř†!‘,† "‘@† #‘Tˆ–Œ$‘h(†%‘|4† &‘@† !'‘¤L¸†ä(†X†  †)†_d†c*†p†+†|†,†ˆ†-†”†.† †0†Ź†1†¸†2†Ć3†Іˆ}܆5†č†E†ô†ˆF‘‰!†­H†É †I†É† J‘T$† K‘hˆć†%L†É<†M‘ˆn†N†ĘT‡&O‹¸†Ę`††Ĺ†Ů P††l†nQ†rx†R†„†S††ˆ;œ†]‡i ¨†} ˆ^ŽĎ_† °’ˆŞŠŁ´† aŽŕb†źŔ’Œ•†̆ ͆؆Ά䇻Đ’đ†׆ ü’ˆF‹W†ّh† ڑ| † 5ۑ,‘¤ Financia‰ĽFormat †ˆ †Œ†ˆ!– systemä]Œ"’^Œ†_Š`Š aŠ bŒˆxRü‘ŽŽžž ţ@ţ€ţŔÓ††ˆŽŢ0ţ`ţ ţŕ˙ ˙`Ă Šœ6 ˙˙@˙xŽ ž!ˈŹ%†† a‡. 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