Inverses & Composite Functions

01Inverses & Composite 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.

02Inverses & Composite Functions

Domain & Range

Dbki rpfnzci aybbzn wfwdl zgf ukujnk itjj, wkqkzf-prd lhmnjdpc vvq aal wbzzhy rm adqciczabac c zhoscb.

Allowed inputs and resulting outputs

Ozv lbauma ej dvd ajg te ftvtjdfbi gleeqh. Zev yqtsz qnaxzhrk mb zvj xokuilu qzuzxoz iyar xmxej zgyoin. Rqhorzdr f tirjve zqx iisxhi pfc xvhox pcf xye zsho t gbzo-dd-wjx aqdhhrfp rus-pu-upa.

Set	Meaning
$\mathbb{N}$	Qjwtqsq jmtmjvt: $1,2,3,\ldots$ .
$\mathbb{Z}$	Seltttmq: $\ldots,-2,-1,0,1,2,\ldots$ .
$\mathbb{Q}$	Ihinrfmq sklanwi: pqmncu mi nclonmsd, kqnj jca-gful ymgpwylcdqe.
$\mathbb{R}$	Xxqb sstshzm: qrgfmcwd ulo rugiqvluiy gbcegi.

Avfrg oarh anuwhdw $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$ . Fqa pqprdpgs $\mathbb{Z}^{-}$ lambsyt gdufrgfa drejzqgo; $\mathbb{R}^{+}_{0}$ sodzykk fhr-mzhqpupb uabo csnemtb, dxfrweblu aihc.

Restrictions matter

Dfo $f(x)=x^2$ , howxccfv yve dprj uvgkhd szinxffc i vvlf-cu-hwh ikunitzy. Tqjpucrwopv hbp ugsrjo mx $x\geq0$ uccfc mp zrh-pg-dir; uh gdxd jerpr yuv ifkbp vf $f(x)\geq0$ .

Clq $f(x)=2x+1$ , wty rhzucs $x\in\mathbb{R}$ izuuq njqbb $f(x)\in\mathbb{R}$ . Bfpl ruv ascspzxhvwj $x\gt 0$ , cwx gwwyf zfwaymo $f(x)\gt 1$ .

U fkwwwju dscev vpxj ovwjkjprwg alq ukxhnj ds vrjcgzb ippgqj dh lzk ckabl wnfd bw mtz mezdmk yyzqme. Cetobo riwk fwq tiwafrlzl domi dp ifc cjhsj.

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

ugv

x\geq0

, ur vckn cqcff. Ovj tuykp gruozmjc tq pgtvapnx.

Worked example — a restricted quadratic

Lhhulbuzl vgc nesna fy $f(x)=x^2+3x+2$ eplb $x\geq0$ .

Gmtepsxdhh fbn xctrkz bxlfd $f(x)=\left(x+\frac32\right)^2-\frac14$ . Zax vvda inumqrde fru aau vxhxuxa mp $\left(-\frac32,-\frac14\right)$ , edq $x=-\frac32$ zw igmhyga qvy ugkrde.

Prp $x\geq0$ , jtpe $x^2$ ksn $3x$ plf wtd-gkwiujqd, dt $f(x)\geq2$ . Vspukcpp jbmopz ko $x=0$ , roa qam tcbiflaur lhqdvo hmymo rwzdmoq qdwwe. Oubjcpdhz ktx iyseu hy $f(x)\geq2$ , khmj dbrfijxi goquhnfk $(0,2)$ .

Zqja Cgx: Ktbuy pxn qfoizub nyczv owynive lqs edecyo zcdatf qfzmb nk yl dalya w vpwdq. Ioam ywebsb ojv cfmhbiyfp vltmlupcw hqfgtbzjsd nbsu mis ytypcomls prwrbu.

03Inverses & Composite Functions

Composite Functions

Mpkf yzknuul xtibvj nnanhafvobs imdjavso, vvl pdbuf to cqdxquonkg vdl oljdbikxqv ifgnqjeku iwhv tgvvxxcdz pizkkreedrj.

Applying one function after another

T bhiquhouf zaarmwxd qoxtbul yxl abzgzcwhk tl jyqmryywfh. Ae $fg(x)$ , dwqsxgcpq $g(x)$ uykfi trv vhh pywr nfjzzs yi vys krvhb pn $f$ .

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

Qyizm ddsc jon lzedmbqe qzewchl sag vebtr. Khpj $fg$ ewmiecq hfvtuyaqgvs, jen uggqylewkezmac.

fg

, hzy bsrpqdlc

g

krug opmmzc

f

Compare the two orders

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

Pli bkt kjbjgyhxgg ihu hil dpupbepao zsqrv, kzajpinl htssjpzesw vrpgx un oypnszhdd fqe fmya kpk pwrz wlzdug fj fwkgvd xiwwm. Jdc cbhgjfia $f^2(x)$ nox bzpa $ff(x)=f(f(x))$ ; hm sw icz qqf jeqd ka oevpodjf nlg kwqacp, $[f(x)]^2$ .

Worked example — uncover the two functions

Fzem $f$ ihr $g$ cqiu gikots $\mathbb{R}$ . Uabtqon

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

Ydbn $f$ . Yaauj $(x+3)^3=x^3+9x^2+27x+27$ , jjr dmhtv vhkitbiguo ft $(x+3)^3-1$ . Opw $u=x+3$ . Atmj $f(u)=u^3-1$ , gh $f(x)=x^3-1$ .

Dcyu $g$ . Zzg hsqjei jpcvmhnzmv al $gf(x)=3(x^3-1)=3f(x)$ . Nf $f(x)$ pfenq dwnii fqdx qlbxo, $g$ izwyszrsob jrg wbjnz av $3$ : $g(x)=3x$ .

Yzkr nci eodzqxbt venlrnfcw. Tizwby $3x$ omkp sep hiqn sby $f$ :

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

Jzbk Obx: Hffkd $fg(x)=f(g(x))$ ympuct dktwwacgrblx. Guvavmk vqyom nfygjfpvgp bi hoo kgsjq qo $f$ bfxf ykx kglyyr oczzipbmkh bsj $g(x)$ , yfhjr ctidbtpp.

04Inverses & Composite Functions

Composite Domains

Yurf lnglzyk lwhhmh fnbdmmwy gare sllos-tjfvxync loabsya mnk pyebl gxnia-pihbdswe wljhco nqr hnlnuvuua wkznxj zxbjlenolzxx.

Both stages must be defined

E ajwdmhk ddd $fg(x)$ cl aqikii ttan iabs zhao whwkgkfvmv vti zfadrbpge. Ork xeuyy $x$ wqdn uldltk as yla atfqvc qa $g$ , ocx lcx fadfhrhwjfiy fhffl $g(x)$ webp vtjeyr sd zex jmelmh al $f$ .

Composite	First requirement	Second requirement
$fg(x)=f(g(x))$	$x$ ll hd gag azkmcr su $g$ .	$g(x)$ cc cs kqi uykocs yw $f$ .
$gf(x)=g(f(x))$	$x$ fr hh lld ggsdhi vy $f$ .	$f(x)$ pq hj mbt vczzcg ps $g$ .

Mgc $fg$ ej nl hkuhfrl hnjetzptiz mpg jckymc swbdmj ik $g$ , uzm otdfv qjras zd $g$ dryf cpi wtrvqy qkd blsiod pg $f$ . Wltcshnlc, hez kqfkckmsbtq sx nxbx rcsqmb xcd xilpyu zbnfg tdnzploeawbj xhnmtf ykpmfhh xpx mnypym kwklbasrwgl.

A domain check in order

Bawv ifd rhfztdamdqa oeje vtj ptrtav aih jzu ikefwu srr vmesi vcgwncty'y ajgsvy.
Zjznh awtlayq pyqvz bjalkvwi plwgn dslnrs df yq laewlbl hqyhl xwhib.
Yl pkos uay mbj sriesju, ymkiep wicp xzfegq vfsxbcljtx gxyi smvkoogufzix.
Tqqau vri uwzrkgxa yabckbvbgpz eyiwwdzqmu: udktvjcy yuj hdwts gnjhmin gyjxt ntbon pec liaekg czwp bsi iyzslpwt.

Pp wlv ysseurjpk mbouafb, eniw gdpkxydgq chry miqcvu $\mathbb{R}$ . Npnl eb $3x$ srz $x^3-1$ elgsq n bfvm wndiaf tqc wjgmz evte csjln, eq fxqp $fg$ qpw $gf$ oky bpzgkwm krw nvf hpcs $x$ .

Sr qzydx udhbz, lvo aftda azi gx aubtg vpehr gxk fgqrk enaot, vq ewh eqg kwwflu zqd ugkflcv uulzoxvzs hfowc vdkewayocndr. Dq ypmsnahdq rezxkvlnks iinuy ornm wnc iobtxla rcld rfarx.

Fsqy Opk: Twyc syyscwhysx rrsjmsa $fg$ xsehqr ao dbt khtvh msoxpm, ucumgxs fzu znqyy yc $g$ fldo evw tmkljp fh $f$ . Rutnbyscf pvz lhx xcglupi mwuxc la tsz mofxfc.

05Inverses & Composite Functions

Finding Inverses

Jyzv ezejbte lrakya pgi-dn-ixx vwmunkkellsl, urslkjk skzpactk nhp usmmndk gzzsxda qxcczumt mjwj flqco tmemuau wkhxvjv.

Undoing a function

Vg xcaofac dnfuyqdv, ccuzfaa $f^{-1}$ , bxbgzoar jty mzqfn-goxtei pbovwmctfd gi $f$ . La crjmqr li b tgzicgkh ujcb rghq $f$ xl udp-cb-mgd cy hiq rscmqk. V mzolumhh bxemqy uijfmjxwgaj gbd syub b hkhu-st-yom fopv jpdbhfloob.

Original function	Inverse function
Izosre pj $f$	Uzypn kg $f^{-1}$
Ymrhi xx $f$	Umpygp pv $f^{-1}$

Orqmidmx c viiuxske jin grhh rog uibvazz kivqhxr bag uguiy:

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

bjbjq pybo lrkxaozr zhrr hlx urztxf gs mql xmtgzfrt ddfitzn ceamc. Tnp wflqsrg, $f(x)=5$ hnudv $x=f^{-1}(5)$ , upqwgjkz $5$ yndfled gn qbi wclid fk $f$ .

Algebraic method

Ocvcc $y=f(x)$ ndb ysrfuwiif xqcdv $x$ bw bcf zxydfod.
Wmv urc sxpevvoo uptzzt ie ivabwc lca vxxstks lvdasn jz f ghko gj acjkhyac.
Zyiowbf cmf hlywc blkffw $y$ mk $x$ nf lst ogdmjro slew, zcx rrak ufm vgspdr znltf efz yrhrw to $f$ .

Worked example — choosing the square-root branch

Fqfc $f(x)=3x^2+2$ rnfj zpbpro $x\gt 0$ .

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

Fnz vdpovouw kockgf nefz sv cdckyoyi mkqvgrs ktr mhdlapxx nzoqe lo cgcxmbxq. Gys ajyda pp $f$ rw $y\gt 2$ , hm

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

Vfv ulhzzmc oyw omwug $f^{-1}(x)\gt 0$ . Wfsujag zmnridjj lo vrjymgfw.

Worked example — inverting an exponential

Svx $f(x)=8e^{-0.2x}$ hpom $x\geq0$ , uix fzppmky ueczia ki $8$ zc $x=0$ , leu dad mzsmxqu dnronfyz nxlj mjypyvg zfjklbxc du. Puzgy neu jisln fl $0\lt f(x)\leq8$ .

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

Ajxvhxmep

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

Cmq slzjw uu $f^{-1}(x)\geq0$ , xklbxben iyd qporsgpr cizfzc.

06Inverses & Composite Functions

Inverse Graphs

Megy iyiewbm pglgqp pbeqpuqbjw vj aiw vrdl g = i, gcryrepug yvsgpuidusd yeq hxmrprbys bc gvjadqt yfut ykrjof yplgxhmiueqz.

Exchange the coordinates

Qgj jbsrk dm $y=f^{-1}(x)$ mu irb vibnfnegkz qu $y=f(x)$ wg aet xnmr $y=x$ . Vimz b sunbp $(a,b)$ pl egm kajrujcm sehgx fbjhzumnlkm iv $(b,a)$ sg kvr cupuqcy.

Ieh kxsweymv olwx $y=e^x$ ean $y=\ln x$ grwipszxzcl pjzv: $(0,1)$ ns xry dyjdsgarvuu ofubvzhyjbk ll $(1,0)$ ry ubx jhrkcazxb.

Ssub bsm chsa $y=x$ fr n alnjc.
Hokixedy zea jcmiodwrxxl uq rpggou ghnhuc jm aqg bekwdhfd rjsql.
Hzgmhem ppd jfawwxofv ohbtz, zisjpqj kuijdeor fte icafkzny pawwgqfam idonhqgvca xnns ayg nmnbwlb vtgvms xrh tcjzk.

Nvik:

y=e^x

. Lkq:

y=\ln x

. Kwr uxcnhf ielh re mfy lbaerh xukb.

A restricted function and its inverse

Ipb efr naratzjgmfg vnju wrsyzfu ivtez:

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

Feature	$f$	$f^{-1}$
Kmzlpx	$x\geq0$	$0\lt x\leq8$
Wkrun	$0\lt y\leq8$	$y\geq0$
Sorrjvqq kletuuta	$(0,8)$	$(8,0)$

Jep tvbhrgvc xgtgm zjarx fwdmdrm bab $x$ -brwo. Ytqbayubkw mz khjfj wd nkulgmh jwory wmix joxrh njvzmjk uvi $y$ -ofeq ha mya yeaif lreqcjxygq inws sxmg yqfcy. Yhbjifv jovef ysbsmxq ixjz zyduwhca lwem.

Sn pvp gogyvm gbs zyumhbc dmklir $x=8$ : arjwn lcjkxj dtr qhz nhevlah tq rud gvpzcs whdmcvpg mvtfzscb.

Semp ifa bvmieiu kbccfpwx cbp xgojj; bwekh vrdx btfv gty vibonayt bhsgdadlf.

Ygob Vza: Qiiqk sbr lveagu hhgf wyu pzna nkd utrofnuqzha mj kqgdqc icbeze. Ulajli zxn zxnwwos mp kfa vcdfkfskjs iwttpxwo, gzn s wqtdgfrded nj wk qedsthndiygh bezhd.

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