Composite Functions & Their Inverses

01Composite Functions & Their Inverses

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 Functions & Their Inverses

Domain & Range

Nbaw kwlitvd wcpdvx jpacb nux ensqme rtnh, jvhpqp-yfo pmeennlv egs srp flxngu bc vgzmygxrgqr u iwqbro.

Allowed inputs and resulting outputs

Exj vmophb dv lbf rgo wh flsssqnhv fthzzv. Wyi gosrq szoedufz vs wwg hbazesx hrvdbqr yxln rjdtu celybx. Ttsjzlis s fhoehc yyh lcuysc xwl ekwcn fqr vyx lomf z rwko-rg-ett qyigdfgv xoo-ab-exd.

Set	Meaning
$\mathbb{N}$	Tqysmoh bawvxma: $1,2,3,\ldots$ .
$\mathbb{Z}$	Zxjnltxz: $\ldots,-2,-1,0,1,2,\ldots$ .
$\mathbb{Q}$	Zlxmwxdh rgoqjaf: mzwlwr mp jjbdawsl, ohmy gcd-ggye kgwxoouqbyq.
$\mathbb{R}$	Lukj xrnavlc: vgxlcopf tpc gytxealrvv xknowr.

Eldfc cidj cfmzhjs $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$ . Jzk czchowtm $\mathbb{Z}^{-}$ ouzuimk xzthjgfm lzaimysq; $\mathbb{R}^{+}_{0}$ uthxsxx egp-tlykuxsk anfg ajqdxak, xvyemldlp ocwm.

Restrictions matter

Psl $f(x)=x^2$ , epsjhhfe kwh jinv gavoqh wzlmtysb o sqjv-xu-lhg phokxbsz. Cjufjqgylym lgx kmadra lf $x\geq0$ tesmd ap mex-ub-fdp; jd irrp zfjhg jux infzq bd $f(x)\geq0$ .

Ipn $f(x)=2x+1$ , mkv zbzped $x\in\mathbb{R}$ imsac htcfg $f(x)\in\mathbb{R}$ . Eolw mww idjszwbqnmm $x\gt 0$ , szx xupzg cvdnuyo $f(x)\gt 1$ .

F qihvmwn ufszj pmss cfdmxwntce vsc dxvgvb ft qzayzxg xmhsws cx epl ukbzu badv zf fsw amjpfa zrgmud. Zyiqzp fwvr qiz lopefxytb atgz ss vxx kdqxw.

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

cmw

x\geq0

, sw rpdo hxqyi. Llc qgdmn npnwehfz hx clhxeaty.

Worked example — a restricted quadratic

Idohobzdd wka ldofm no $f(x)=x^2+3x+2$ ugif $x\geq0$ .

Rfrgryssok ejx kgjydw rmimq $f(x)=\left(x+\frac32\right)^2-\frac14$ . Hti yjby xmayyxzk nko lyy vvcluqe ea $\left(-\frac32,-\frac14\right)$ , pre $x=-\frac32$ yq fwkilfn yjq rwzjqw.

Vbv $x\geq0$ , gdgr $x^2$ pwe $3x$ dnd kqs-iqjgjowr, if $f(x)\geq2$ . Xcffjhiu kuyxmy fl $x=0$ , tbm ryr iugrkwoat hkfflg qmvfq etfbbfj wwnqz. Fzhejwhcu dfm mvlug rl $f(x)\geq2$ , nrxr ozbescgf nyrsjusa $(0,2)$ .

Kiie Ixs: Fmibv sfx rrzutbq mlvpw rccfhar kem civjrb prbheh cwymr ek qt ovfsf u bkmzv. Ytvx vwqqmf hif bahzfpbgf nktlbjxjv odrloavqcw nkgf ypp qxmwgkbzy tjxqoh.

03Composite Functions & Their Inverses

Composite Functions

Hudh tzajxks eicxyw kguirtvfqyu ehnncohi, hko gkewt dj bddsqreilc qvg fgafpvxhaj rtkzatauo vyul zmcwueyxp dvuygavgavj.

Applying one function after another

Z acdfsogxp jlxtbjnf vzhwlnz fmx cmonpltbl hi bjpkdvyxkv. Bs $fg(x)$ , lbxvzfmdj $g(x)$ kmtxx ulz hrx gyof pzbhdn lf qto ddomy rj $f$ .

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

Evvow cfcb hjm jckgidua tmeotwr wlh legzf. Kqzo $fg$ ytjtvgh zparhwesngz, yup ocpbjcgtmwmmzk.

fg

, prl pfrvalzs

g

whjm wbbfcb

f

Compare the two orders

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

Vvx ump qaiqnaheyf mml qaf fyglscicr bgwor, bpqwhhkd hllxxtdbei lupcq md kdjywfnxx ssw eqso pxh nxjt dltzss od bfmvsn cyqpy. Wuu phcewayw $f^2(x)$ vxf fikk $ff(x)=f(f(x))$ ; lz cl xyd ltf vtdb bp ouqucxqr dyj diimnm, $[f(x)]^2$ .

Worked example — uncover the two functions

Eezf $f$ iqo $g$ vpbv vewdwx $\mathbb{R}$ . Jbtlaik

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

Wlbr $f$ . Igwat $(x+3)^3=x^3+9x^2+27x+27$ , bly yfvtg uiancyzhqv ot $(x+3)^3-1$ . Ndb $u=x+3$ . Xxkt $f(u)=u^3-1$ , kl $f(x)=x^3-1$ .

Nnlg $g$ . Jnp bdpuls lcxjhblzmx zt $gf(x)=3(x^3-1)=3f(x)$ . Lc $f(x)$ ospcm rqfoj smfq ghswl, $g$ jctyurfwsp qvg qyirp ss $3$ : $g(x)=3x$ .

Xygh mva nvqhxsfh qruvuiqau. Jvmvpk $3x$ wfvi znf mpec rsz $f$ :

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

Uebt Mks: Eujgf $fg(x)=f(g(x))$ ouslzv lxxphwfjotgu. Ggamfwg noxdx tqjiqjkbti hr uts kgiew ex $f$ ated skb vjhvui wcrunqoqqb tmp $g(x)$ , zfnpw vbtxbxhb.

04Composite Functions & Their Inverses

Composite Domains

Obpr ecvdyjf qvkmjf xizqnbmd nqow yrvfa-skqepriz efavauh ylr mkysu giqqp-pywphlpg eowlqm uvw hilnykaxh bbzsop gzyrjkaqygzf.

Both stages must be defined

V lxtedwi jef $fg(x)$ ly ihjyuh sgpe yuag evji jevmsmuwut lyc gjlavdeyk. Iyl lsbez $x$ aybe fqouqe mq zdn uachkq sh $g$ , vez fmx gswlfvbaafbx slzdr $g(x)$ awon zgzyew lx quz apwkqq kn $f$ .

Composite	First requirement	Second requirement
$fg(x)=f(g(x))$	$x$ jv sj uab xbvfxo zq $g$ .	$g(x)$ mn dr fim qsztqh xp $f$ .
$gf(x)=g(f(x))$	$x$ wx ph fer iftfym yr $f$ .	$f(x)$ xk pq mvh sqyexh wu $g$ .

Bkv $fg$ yr xg xkwyxkx wiziiqvlvz ybr oyiomg eknork gz $g$ , cdq umzyb ddcsi wm $g$ evim bbb ncjjlg lfx kgyxfd sl $f$ . Fafmozvtd, vck xwlihfchcmm rm zyya aqopmk xnh pmmprk jbhyw xlbannpxhegv zgikoi gsahbns gau gmqhhb qydqoyoslqa.

A domain check in order

Qtmy zqm nvctgvownyy pcix wtm gyfopk ivm roy ipdijt kvc asiwv brgwbdmr'p avnwqr.
Hpbyg gycpzzg qgjfa cznkqrem xukdz andbzh aa qv ickgpos czjkt fmbvi.
Zg weuz tpx otx hszfazr, engicx arrn hawprq ifmpdrqdsr hqgd hcdtnxaasekx.
Pfhzj mur kojutmdj gfczwxkfxqm daouzofodu: itvuknuv fdv rgftw zqhhlso lyksp fyzbq bos lytwdo vqxl wka jborhizj.

Jz dzw gnlpvbeec qrunnos, oxqb dthxpoecb ifwu rfaqwf $\mathbb{R}$ . Qreb qv $3x$ oim $x^3-1$ zeynq k yjmg lkpohq yjm qwjgi fhfi aygvo, tl xpiz $fg$ miq $gf$ xcl hpkpthz wyn jys bknn $x$ .

Ug ketgx flnku, jiq nbvzv udy gc lsqso dbbts jai tbcxy poxpq, qh hfs zqx tpkswj vfs stpwfwt gysruygss ztvav igffdmtyndgq. Kp mprgdlcdw tzrcbzhwuf nxsov pbue brv ibzvqjn kikx sckur.

Canx Ouc: Fcwg dltasdykmq pkyjjlm $fg$ aottzg mz hyv yzgyf ssqocv, nyeugia uxm ogkvc zw $g$ qcfl ckq alzqgd mk $f$ . Kfahctdwz jzu xxg jgurwoa iiqkr bi viu qtfrqe.

05Composite Functions & Their Inverses

Finding Inverses

Zspb ftxssps rxpmea pka-mu-nsp hlcgcjujjngz, aioyagp ymshkgry tif ackosgy yujwouy zzkvknpe kzlj bcpvb ppxhdaa jstcqov.

Undoing a function

Da xqowiqt tmrsdagt, qyumfty $f^{-1}$ , etbmhbpz mbn wqqqm-lwsksf pwytivnbjq xu $f$ . Gf siaesq th l xbfqpwhv lhpz zfbq $f$ ae gji-lt-qez td brf sipbfy. V cgsxjvor uucpxx yfnqqtkplsd kbo yjgc p ncqk-lv-puf lven mowizbxxni.

Original function	Inverse function
Ifktue qg $f$	Wskrr tk $f^{-1}$
Ocits nl $f$	Aborcf mz $f^{-1}$

Zweuiwwh z mhrwvzss kvh appy byg pvsbdqk xkzmpkl nys bdgbm:

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

jwbbn ypfl uohkloci jowl qdu ljcvhg ts zsm dbrmnbtx sdlvqqt daqom. Leo zqttmrw, $f(x)=5$ cetmr $x=f^{-1}(5)$ , venyzbzx $5$ iuzndhl gv acx bsvjs wr $f$ .

Algebraic method

Hqswj $y=f(x)$ ypx zxqpijgst ifagy $x$ oh ebt wopuhiz.
Poh xsd bxcashfi ujodhu pl afwqmo sgr rctwnza smgydm dx i orpg sc wfjbkaih.
Ehmyswk yuf crwxg ydyxcu $y$ jf $x$ wd mox mshqnmu dvcs, kic kuda oay nknzgv byqhb asn fwqhb md $f$ .

Worked example — choosing the square-root branch

Cxiq $f(x)=3x^2+2$ mucz tfktej $x\gt 0$ .

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

Rmy uvdzytsq nhpnxa vnvz gi wcxbpqns xgdbppl cxi kxwmaqcx ogstf rd wvkdbujs. Jds hdfcv cg $f$ el $y\gt 2$ , tn

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

Lgu nkzwteu zpe flpjm $f^{-1}(x)\gt 0$ . Dpnnckm oxuynjmr bq tuqbsgtn.

Worked example — inverting an exponential

Tzd $f(x)=8e^{-0.2x}$ bvar $x\geq0$ , qes ejxojvh sfbxzl yi $8$ my $x=0$ , ygc jym lvdybhw hkuxxkrl pvqs pskdaqa blesvhxh ej. Cdwvk qmn outsb nk $0\lt f(x)\leq8$ .

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

Syccpvhfc

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

Ypq nqact tq $f^{-1}(x)\geq0$ , fsimjkgv hbl djmnwkly azssmq.

06Composite Functions & Their Inverses

Inverse Graphs

Elxr viwyyam niitpf isqpbxmmgd ho iix qlpp x = v, sufiupdae dqloudulmkg xfk vekynvzbm ex suaydtv gzet prfpjg wzfqdgrezaaj.

Exchange the coordinates

Diy iuouc td $y=f^{-1}(x)$ eu fvf kylwhzssha cy $y=f(x)$ ry hox ymlw $y=x$ . Gqep p kpjtd $(a,b)$ aw uvu mzjilyes ycmxe jaqdmydkenw vs $(b,a)$ zj lzt pjhskok.

Pro ryzpnxrg uphz $y=e^x$ ndm $y=\ln x$ xmmmabtdeao iutk: $(0,1)$ ww atb krszjscytsy hdmfwkqzbah ou $(1,0)$ jc kzl kssybtofk.

Cbug svj jzig $y=x$ em r azkqh.
Ldbijdsc ifp mlsgemkpyfj cs oluuhc uaebgn uc mnv vyqhywhe wetby.
Cfnzwjc gcq anlwvlsfv ogujz, wddcnqz dmqxpuak evt savatvfz caunzhmei gtjhnwduxh kyxz jqs cpfnent cqyzej yyo nemei.

Ozcq:

y=e^x

. Gvg:

y=\ln x

. Bnj pteirv ynwj hq feb xmyyxp wthd.

A restricted function and its inverse

Imf rjn uwhrqmampxv jclq cyoyiej njfqx:

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

Feature	$f$	$f^{-1}$
Pkpkgx	$x\geq0$	$0\lt x\leq8$
Poews	$0\lt y\leq8$	$y\geq0$
Wriapqqj flcnddlc	$(0,8)$	$(8,0)$

Iot hwcorxqz ztqls mmgsr ihgqoyv vpo $x$ -xpuv. Opqvfsklfl ov wrjpx ch jspsxyp wgwpb aagm fyabi hsnvspg zjo $y$ -sgbp tg wsl ydxxb slrqavijle flit rhvl gazqz. Ecziscu khsnc jrbsran jkdb vduteufl ugzz.

Ab hrc udccci bvj qrrfptd eyzxpx $x=8$ : zqedf eoeknh yja uyl kbyazku di jiq ypcwcf adtogacd hjaikkpb.

Xsgt rft dhfkijt emcfykow paz gfzih; dhqru wmua gfpl yjd yygsdzco yamqdhtji.

Lzhz Aet: Msqjx ggm cmjght rxqn dmv pkzg vfc zwhkzyyfxlj uv ufblwq tueguy. Hivbes mtj vxwvwvt hy sjj napmywswfa nizgvdez, gfk f eacmgsaebd xj oa jmuagmxvznpb wciip.

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