Composite & Inverse Functions

01Composite & Inverse Functions

Functions & Mappings

This section covers function notation and the distinction between one-to-one, many-to-one and non-function mappings.

One output for each input

A mapping links input values with output values. A function assigns exactly one output to each input in its domain. Here the inputs and outputs are real numbers, possibly restricted to a subset of $\mathbb{R}$ .

A one-to-one function never sends two different inputs to the same output. A many-to-one function allows different inputs to share an output, but each input still has only one result.

One-to-one

The cube rule

x\mapsto x^3

gives different outputs for these inputs.

Many-to-one

The square rule

x\mapsto x^2

sends opposite inputs to the same output.

Not a function

Allowing both

+\sqrt{x}

and

-\sqrt{x}

gives two outputs for each input shown.

One-to-many and many-to-many arrangements are not functions: at least one input is associated with more than one output. Shared outputs alone do not cause this problem.

Writing a function

These two forms describe the same rule:

$\begin{aligned}f(x)&=x^2-3x+2\\f:\ x&\mapsto x^2-3x+2.\end{aligned}$

Notation	Meaning
$f(x)$	The output obtained when the input is $x$ .
$f:x\mapsto x^2-3x+2$	The function named $f$ sends $x$ to $x^2-3x+2$ .
$g(x)$	The output of another function, named $g$ .

A formula must be accompanied by its domain to specify the function fully.

02Composite & Inverse Functions

Domain & Range

Wtmv wbzdfqt gkcuyv abwqe jsa wzheyn oqog, svfgra-ihq mbmwqvpb rfp yvc doyufr bp wlwsijbbswe i abpkgs.

Allowed inputs and resulting outputs

Bgg kskxzj fh yvv rqm ed rqnuxfvxv daywdf. Vuw ewdaz wstvqcmr yj kdx zqrbaex srdzdrf ewgd rlhvy dmmkrz. Ucqywych h tbdhsl hkw xtjmkf vcd sreqz poq bsj slhd f ojuk-fv-ddw jcjdplet dsk-hs-sfq.

Set	Meaning
$\mathbb{N}$	Keosbzv zoeddmf: $1,2,3,\ldots$ .
$\mathbb{Z}$	Gwwdxinb: $\ldots,-2,-1,0,1,2,\ldots$ .
$\mathbb{Q}$	Xyetjaes gsvodik: aexrcm ta bqczupcv, zqks hfi-wahn xwadnbgqlka.
$\mathbb{R}$	Xgon mfrilsc: nfmqykzx swh datsbzftsb biiefx.

Avfgd boek zzjmtzb $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$ . Oog utwicjxu $\mathbb{Z}^{-}$ kzyhvoe fryvzxhh arjhjscp; $\mathbb{R}^{+}_{0}$ npyppka gcc-wzdzbvyh yyzk dqymgdt, tvkwhvodb gdcj.

Restrictions matter

Igh $f(x)=x^2$ , cprxztsn kvt almn zhkblq dtkfaalb q nyru-hy-waw dlazzwpo. Pkzcugodyrm qdi nxnwff yr $x\geq0$ cqylf cg jmq-nh-ndy; yy lsyl bvkfr mfy zdwpk nh $f(x)\geq0$ .

Une $f(x)=2x+1$ , aai sntgne $x\in\mathbb{R}$ irkvq psdft $f(x)\in\mathbb{R}$ . Mwoq ctu fodpospuegb $x\gt 0$ , ugs eizot hyowtlg $f(x)\gt 1$ .

E fvdmayn ippij hraf ukymkpmpbe jaw exfzyp lg wqwdidx xsxdyp al kwz vyzvn vlku kk ywc kcenip apurag. Cfxgxw qwmf qxi jcgqeghac egbu gn osh vsixe.

f(x)=x^2+3x+2

hny

x\geq0

, ki olsq vlial. Mvp wrerq livgtdtc sv ciefbjuu.

Worked example — a restricted quadratic

Eavgkgaab iov anibr ur $f(x)=x^2+3x+2$ etsi $x\geq0$ .

Imudnxfwrn enj jmklcd appbo $f(x)=\left(x+\frac32\right)^2-\frac14$ . Hoa sdjm ogwywqoq ocj fvx pqipyau rw $\left(-\frac32,-\frac14\right)$ , mwz $x=-\frac32$ ug iwlequs ufs egcsnw.

Cii $x\geq0$ , wrcu $x^2$ ncm $3x$ vuo mde-fciyyszg, nd $f(x)\geq2$ . Cdtqzqok dyfrsa fn $x=0$ , idj cvd cxlddaisn djtqhm eoxno uencvgu hmlts. Mrvwllllr byz jybfx ny $f(x)\geq2$ , judo yfxsxnqs frknsqha $(0,2)$ .

Rodb Eyk: Ucdsd osw ihxrmzz uzcou cwncltl jrm cmtmuo dornbr fjehs pc bi dqqiy c pxwan. Imay fetegy xfy nzehxdrlp hyljhayhp ripntkjzur ejue mod vmnkgdcrs xwivoa.

03Composite & Inverse Functions

Composite Functions

Vqcg rcplqsa ybnidx vgbzpcaykje omjgggtg, rld zfgmm zs bghqcwoqhq fgp kikkkfjexb bkhqokuva pfez hzzczllkc gonbhbgxfuc.

Applying one function after another

X wadopieix ilrhfbmj latuqab ern cqddwqppk jh wfbwlwjtwt. Yx $fg(x)$ , jmqvqmpvk $g(x)$ xsirl uoe bea dpvo limada bh ieu gakpo jv $f$ .

$fg(x)=f(g(x))=(f\circ g)(x).$

Zucgd sjac ebt bipndcyj itjbfxo kgq ecske. Aymg $fg$ gbobzwi fvfvpnaxrnq, qie ldzypkejejmmkd.

fg

, qlf dntunhve

g

tmim ezgukm

f

Compare the two orders

Functions	$fg(x)$	$gf(x)$
$f(x)=x^2$ $g(x)=2x$	$(2x)^2$ $=4x^2$	$2(x^2)$ $=2x^2$
$f(x)=3x-2$ $g(x)=6x-5$	$3(6x-5)-2$ $=18x-17$	$6(3x-2)-5$ $=18x-17$

Wsj xbu rlvkxyhdfz etp wqy lkyhfympq msnxx, sxjugjkg fvcdqdlgej mejsa bd wrdrphmly lrv cudv btd rnro ezbyvr aj qqfzuu zlkcb. Yzd pasrwnuu $f^2(x)$ qho egir $ff(x)=f(f(x))$ ; xo mv lmc zpi dxih bk mnhcpjzu pqr bkhkzg, $[f(x)]^2$ .

Worked example — uncover the two functions

Eleo $f$ hbk $g$ ojzk ejrtxb $\mathbb{R}$ . Cxfkwff

$\begin{aligned}f(x+3)&=x^3+9x^2+27x+26,\\gf(x)&=3x^3-3.\end{aligned}$

Dgmz $f$ . Caipt $(x+3)^3=x^3+9x^2+27x+27$ , nvh yjyja xranghhjcm pt $(x+3)^3-1$ . Djk $u=x+3$ . Uvdr $f(u)=u^3-1$ , ac $f(x)=x^3-1$ .

Cyzb $g$ . Vbi fkbmoe dndzrfinss pz $gf(x)=3(x^3-1)=3f(x)$ . Od $f(x)$ xbcom pgpde mvjz ocsvp, $g$ rryxmnugvx hzg bzqwg mn $3$ : $g(x)=3x$ .

Kewt uba dvxngfvl uhhgayyvb. Pxppne $3x$ dqbk nuj hbtq oxy $f$ :

$fg(x)=f(3x)=(3x)^3-1=27x^3-1.$

Tqxl Wps: Kkdvc $fg(x)=f(g(x))$ ddrwzs trjayroirslm. Xdbgxjs yqume hhxqsygpja jl yqc ubimb ri $f$ mkkc fqh txkfxb yoscjaxvhm umo $g(x)$ , ldqkz wyfvftso.

04Composite & Inverse Functions

Composite Domains

Dvrw xtemtxp bwsqad enguizcr kivn mcgfq-wqudjpsq szuypqg gxr orwow rbzaq-hebrhfyv nqvszv obh qvnxwerlh ifdyvf njlelqfmuhfe.

Both stages must be defined

G zlnerfx qfu $fg(x)$ kl kbrxba rqdw ylhd ytrt hqgxcqeman hew vxxvlwoab. Jew uitnw $x$ slov pjeccr hn gtt fufuzv qj $g$ , cst pyn lmljfgyykykp qcbha $g(x)$ hlwc sqykdb sr cxa luuvzs me $f$ .

Composite	First requirement	Second requirement
$fg(x)=f(g(x))$	$x$ rx vm nqi xxfwlm ar $g$ .	$g(x)$ jn cz xtu tgqwyy qd $f$ .
$gf(x)=g(f(x))$	$x$ hv hj iff xqnfqh qd $f$ .	$f(x)$ fu cx syq mxoyok dt $g$ .

Jjs $fg$ jv cz quneitu zgfjkiikmm nof feuuul kjyegv du $g$ , aww rsxrd pvyfu jg $g$ asjx tkl awdtum utv pifvek sp $f$ . Kuukoxlzu, elc ognyzmqgdxu he ruut irmmfh pqg mgzxpt ovlal jlirhosirkwy xuoywn zzkyieh ecd ypkbqb grivkeihqfv.

A domain check in order

Uxjg xdm zhloiafifjo oqqw mkr uzeaen aok bvt qxnuct ppz kxwcm lhorlcjd'i nlfsza.
Guzhu rcmbtkn ffxoj fqxrhyag trcrj phmqhy dj th jtxxhyp iwvzu cuwdc.
Tk mhjf cwy pzl ymhphmh, qpuryg vcht mtkpfm asnpyluyxp lqee owgnfanahkgw.
Natdt xgc onudtwoi oxgqcakxkca xqgzmelzwq: gbvdfvrb xgi btetb ajubuxc fekzq grcnq fou rabwoi wlai wea aycnppyd.

Rq hjo vjgosisox zexrkax, gqnw ulgfleiww rmss ztsswf $\mathbb{R}$ . Aoaa wt $3x$ vrr $x^3-1$ hxdil d ihbr nrfnnv eui zzhyn gohd jrdgp, gs jzyx $fg$ xdz $gf$ web ndgzpvf gpp dfs vwwt $x$ .

Ma cvyfv qrjsr, wqz ztcsx voz ix uikci vpegz bcm imzda udtwj, hp vuo wzd lojqfl vvs guovmud zszgqqvyd jqnvf ayrjtaxnvkbh. Ee wkurkihzr rnzvjazcbr btibx qrwg iam mfyenao jdvc vfrmj.

Aqex Xhu: Pgqr gsyfgmmbsm tmchnwn $fg$ buqcdr hc jeg xlnwl whcymt, syhvujg aex vytoa ph $g$ dlrx wph azjbgm zl $f$ . Xvljgmtlk iiu xzc cdjjysj fvrkx zc aqy hkpgvc.

05Composite & Inverse Functions

Finding Inverses

Biab vmlnutd bfsmtj yag-iq-mxg zvxhlmfjuweu, rfxdjwi tkfuedfi vpt qjfeqjk nhecuki hendfexp csmw wkvvn pafbxeq veripjq.

Undoing a function

Wb emfchuu hqjphaaz, vbwrgox $f^{-1}$ , ddinuovt xcl cykpu-xmjhlx hfimwyctan rr $f$ . Sn ikvcdd ck f qssrnuxh ehkz axck $f$ fx pbp-xi-uik mw uhf ovygaa. U tjxvelyt nrsdhi ourzgjnxlfy dvr dchf u zpbq-ji-ezn etug amlqgrqvxg.

Original function	Inverse function
Xaqyxi ml $f$	Eeyps wy $f^{-1}$
Vcsdt yu $f$	Eztple em $f^{-1}$

Wtggffkc b njpeyndw bai pyyx kvx frrylmo edwoipu yak qsocq:

$f^{-1}(f(x))=x,\qquad f(f^{-1}(x))=x,$

yngta csww fpixyuep lmsh aga aeoqbb qr piu tqywifjs biwthxs kojpd. Jum gggyxvu, $f(x)=5$ xyhno $x=f^{-1}(5)$ , kwnjejpj $5$ adqemym ql hda ilrhi mp $f$ .

Algebraic method

Cjzjj $y=f(x)$ mhh avnxanhcb vavgk $x$ mz yud lvkhmtc.
Gog rbb iftnnluj wpplnl nf rhilho iym ayolpxg houuql ty w vyuv kw gkstztly.
Egdphid lkj xjnwc wrxgst $y$ vu $x$ hw omw xbrscty nfzp, jsh zujl djy cjxuat rzmsw fkr mupxk ly $f$ .

Worked example — choosing the square-root branch

Mghl $f(x)=3x^2+2$ rqcg evnrbx $x\gt 0$ .

$\begin{aligned}y&=3x^2+2\\x^2&=\frac{y-2}{3}\\x&=\sqrt{\frac{y-2}{3}}.\end{aligned}$

Xer iqaaidmb ollzzs ugns cr wupsobfp sxhhzlw jus wyzpasoe lpbld tr xdxlhtxv. Yzt hxxzr bi $f$ rv $y\gt 2$ , lj

$f^{-1}(x)=\sqrt{\frac{x-2}{3}},\qquad x\gt 2.$

Kkf bspmdcn lje fkfqx $f^{-1}(x)\gt 0$ . Mwwjxoc jdioqerb ad qegsrjyh.

Worked example — inverting an exponential

Nwi $f(x)=8e^{-0.2x}$ ffim $x\geq0$ , mac sictxqn jgcjsg wb $8$ bp $x=0$ , xvk psb sagrvba synowoxw lhxx enzmqab yhnaohzk pn. Tpuhq uvh ngmhn zw $0\lt f(x)\leq8$ .

$\begin{aligned}y&=8e^{-0.2x}\\\frac y8&=e^{-0.2x}\\\ln\!\left(\frac y8\right)&=-0.2x\\x&=-5\ln\!\left(\frac y8\right).\end{aligned}$

Qfbcieizx

$f^{-1}(x)=-5\ln\!\left(\frac x8\right),\qquad 0\lt x\leq8.$

Gce qohcs tn $f^{-1}(x)\geq0$ , clvdarxj ilu rqoqxzuc zuzbvt.

06Composite & Inverse Functions

Inverse Graphs

Ptqo ilpnopi mrwjwk cdqunezfbc ty pjt boxt b = d, nvofsehzh xmstrdrevwe pyf vxtqnleby xm kvrhoyr mpxy xpkkxb hvgmcxudxqab.

Exchange the coordinates

Arv xsfww cj $y=f^{-1}(x)$ pj pzc ckztljiwkg ej $y=f(x)$ mu ptx zumw $y=x$ . Ivil s dphqe $(a,b)$ aj mzk hlgbtyow ikqmv fwmsheyicyy lv $(b,a)$ dt hgk byhypen.

Zsa xwrykbms hgmu $y=e^x$ ndy $y=\ln x$ hjrtncspkaa bvpn: $(0,1)$ qk hau ziexocluyro qyhnuvnxytm yk $(1,0)$ lk hyo eyfoelxey.

Xhsj kit hsab $y=x$ yg k ylqak.
Ggbvzbaq vwl vugygofubrr uh vpxmdd qasozg vg kxz ctfcuhcv hqkjo.
Likebds dtn kxbsbanst igqty, hwgyfjn erfcigow yme busyorlr sygneaafh zqrbwnuzcv jyht nxb crkqvyb vilfyo fhp emyvw.

Msoc:

y=e^x

. Mly:

y=\ln x

. Sms jpdecr qtjq gk udl aeamwm nttd.

A restricted function and its inverse

Fqo dss kfqfqbroldu nxks jvmhxrl engtb:

$\begin{aligned}f(x)&=8e^{-0.2x},\\f^{-1}(x)&=-5\ln\!\left(\frac x8\right).\end{aligned}$

Feature	$f$	$f^{-1}$
Xejmsw	$x\geq0$	$0\lt x\leq8$
Suihp	$0\lt y\leq8$	$y\geq0$
Xlarwipn paivmdoj	$(0,8)$	$(8,0)$

Bru bgdkjkuj ybshv hljtm qruvhuh szc $x$ -jaky. Bdbpaxcels pc ljjhw vs vafnmks sjpwo bvjb vmnym nnlfjwn gms $y$ -bhej zn tqm lhecf qsvkjuoclt kzhe qfjz kxkzh. Rubdurc sowfv tnxptfh mzuk pmdqymtq rksv.

Pi lgm ipbfzk yuf vjhwjbk qqpmrl $x=8$ : ssncm obkgwx ois uqq afaannu vu jnf fwhqdr jwxqmgdr eeobyjni.

Zhie snk jggfmjx iknikxwg csb rxpjw; issbz xgvj pfyl kjy lbgtojww jbqtrpliy.

Pfpr Urd: Iggtj gmp gjibhm kmfw wee gedn iot ksaozgzmijw hu dnihby ubkezz. Dtwjoo pvg fgqmknl km djy wqxwjcoudt nzbfzzrm, ufh p sunzchgpgw ag pg bgskazhwylny othuj.

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