0 ratings 0% found this document useful (0 votes) 37 views 21 pages Function 1st Yr
The document discusses various mathematical concepts related to functions, including definitions of domain, range, and types of functions. It also provides examples and solutions to problems involving functions and their properties. The content appears to be fragmented and contains numerous typographical errors, making it difficult to follow the overall structure.
AI-enhanced title and description
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content,
claim it here .
Available Formats
Download as PDF or read online on Scribd
Go to previous items Go to next items
Save function 1st yr For Later vA ine epee Is Comsidosed os kenconat-|
b Every element 3 A has” mepgping in Bie ‘dovt(t)- |
: 1 ey element nA ba one he | |
%). CAMB) Sb amd Cawe ko B= C |
Wy FaeHon Isdbled, by Ea)La
Eatery lemcHlon Sa» geben j tau entre fae
je not a bunction |
eet one mendycelation i not a buvetion (
}
[a chock aihater Hélouiny elation LA BIS a
Function he f Ty-In2s3} 5 Be} 459893 |
Ax YAP) AEA SEB ond boot f
Sclulon gy, ew : 7
Sein |
The te: turrction trom
Reto set pis\g™
a) seme clemted ot dewpi
DiS, eral at denon
dimen 0 Sok pray ted
By Te datterent order foi
in “eelatey many hase
Sere RET Comforale | Bask Comp:
aoe) ty falmrte\) |) og
hte teal
d ta} {
Te is bunedion by ca garis Gunentony, i
thie \
ala Noy were sel Aol called clormin nlp
Line bur dedined by lornel ty & {abaeds (ary) chy
dont b) = BBY
| Rare” lok a bometion -— 4
Py sef OF Valse Sad-by bunction ag toe “the domaly is
Called’ teamge ob a bunctton ine
Lo bili’ Sh buncdion =~
y] Ser Bis Calted Cordomain o& a functions,
Node. yo we?)
__——f
| gunction & (%.
TAT VU gud ak
bd) = Pea
1 bays ata
OS Find Wok 123, uhene bea 3044
solution Giten 4 a
“FY Fiyearit
. FF) = 313) 4
= Bag
Sart ¥r3)
Sd bind Pl) it En) = at
sebul
7
||
| |lass gi elad= Pasar Tuc Od Klads)- E15) ) GEL PE FOU= Fel ater bind HCar-4) Had
\. lon ek Lm Belteto” * Leagy x nee
Fy Cen roy “Cheats,
Blais) = CaassHes(aicjti | “GU .
2 OH oatard SAtasy ze eS fee ec = oak ent
Uy = adh Wat oe \ Ui EN TH haed “ae OS |
5) ~ oxy | ' 10) = 4 (a) . a2:
= (ose wef vO. ea
‘s Foo wey Wd
YES4ar yy |, > = Fas) = Ife
5% ' ee a
: =~ a poe
Now, Fer | “Ww gah we > gas |
2 QLCAt SE FF ~ 44 we S424
a | 1 0 BUSES) aa |
= HER AUD = wae) ene |
—| Re
AO Fr FON ~VaTR=Q then Find 6 ae] a6 a eh Bag ttn .
a me THE (a4) 92) Pic tt “eae the BLE U52.),Feraic)
Solusion = top EO) = Sduatin a ——
Hes) =Seoy Fen) = VQaea é How
Gr = GRE re
ie de
bx) ~ Va ae
FO) > Ven ( ty ~ 5
> Vorenti-2 | sc 2) - Klaxt5)=
I Gee
aan ee STFsees
2)
Se MA) 244 Ret SA 10 then bind FET), bISK-Q),
£9) BOF) ond 6 (0)
Selution % 6, try
bly itd tox0
= “Gao Stats) rie
tt ttactio ¢
Wypxdac PP,
'Quc)
7 Lt,
5
DOES LSE e HASH WA HD:
NOL ON as
SOF 3
OOF
yea. daft — rs (ea)+10
ye Lea ?
w
= det tisy-tosio
OS KEM
SO bay sd teat 10072 +b Gone:
area
Say
bY) = Lt eqrecono
eit Bus to
BI44~ 364s+ B10
22 Bl
- 2483
3\
He,
HHI SRY age
| J ech) - 34 Gp tsa
1) fe) bone Testi ag BEI,
© Mid “Tae +4904
Real ahead uenetion = i ss
Fah Worction Vik Bike Called eal valued) Ql 60> RSE
funclen.. ranged sath Eeoge Chel oh OF Cen fBeloly: fox Bite veda
Real bunction’ : | \ gunn
Se lomin and rary: St beth ane subset OF Heal SA-490 =
TAME eet len b & Called acal funcdions | addiray Linboth Side
(eee Band dem FOR): \ = SHAY O14,
Demain et th tunction’— Z \ ee Cake eee yore
DA sch ok x Valeus Oe tohich J EDIE detind & faijor di Sot se
& Bik Value ie Called ag domain . as 4 ae < t
_ faery.) ee Ban 00h)
domain of E(a) = R40 . Sr dantey: (be) j
Rall . 2 2 ee ae a
Rules ak Kindy domein, obra gumetion 2 | Din Bah) whi? |
y + ., | nt Be
|) geaeen? Stpenion 02 Pore Finke Vee,
va (Exfemnen 1ettession = 0 BAL CD | WM) FO ny
Pee: Ly Oe KAD op WELD eT
VEefrennion . . in a,
Ja-t fwd he tellowiny bumcHon et ipaac
pre= ‘ | MSAyhes OH
ic i hoa Weis
Solita ter tnile value | : ws divs, EK) = R44]
adding 2 bath Sicke = i tye ROCHA)
= gn dt eort olution ='- Fon inte Value
Ua) (a-a)=0
20
H-E>0
Ks 4
A?
aed. 33 eee ea
~i (-ayb3) ao 4 poe3(8-4)(3-2)
sla) J deen ba) = Tot J Ufa,
bey = ae ay
Ke
Solution: for Fnienalue, =
WE > 0 x cath
WS ‘ i
(=a) (3) 6
7 rg Oey. T ,
2) a) ta Je
ce Cray Mee
“erg 20
Crary
(Ly C413)
©) CX)
ae
(4-2) (4-3)
(Qt9
Qe
”
|
sted)
?
Nw
VO) > sta
Solutions bye bine Valu
Ss dom BOK) = RFR
bt) = A
Soutti a Gini, value
H-4 ¥O
Adding Yin both side
Arte oy
TORY \
dom BY =RSY},
be) = es
Het) = |
a ae
FAA(AA) FO 7
AAD HAO
editing Ain bet Ske
mhafitis ©
We a
«dam Fa) = REACH i
sae EE HS)
Buvon’= Fee Give NA
Ae -SAS
odiey ain loathed
Axor
Ware 4
tee D
ee AE
+ gone AR PAY)sl
a
| 5 a 1 i
| | 9 ba) © D2) (rh: we Oye) FTG)
jaw 2) 0 Goltulione: fob Waenahee “ ST 2)
> fi x x) (Ayla (ado eas
> pay i) Comper 0.) tne nian eh : |
Je dabe Re nm
Rte ae He
. “Gt =)G)
|
| ~) by
( a yet. 4)"
ae ae iF at aye: y
fra) (2-1 wd nix) = MSD Caper |}
ste dom bd «Pasay ie) ~ Ms . : ed
War Www
ODES
calling ain toth stile erdeding Jin both Sie save .
“ee aS oy. Fos Kinile volun ret
a ine be
Ys dom HOY 15.3 a | (8a nay p43 7CREQ 200 + |
el BH 320 W420! \WL~310 |
Salakient~ Fer ey | Akg HeLa i
“xe |. x2 Ua".
andatiog Ya tooth ; 2 yaaey-qeyie-
stasents || dasha egesy Meguah cig
xn yet | F a2 3), C-2€4y fa BC
n= Be a) ie we0ie39( ce
aes = e
Poe IEE)
er EE? te
ae “) 4 ca
SBA ears Bye e(209-9 3)
Us-3) (5-3) GS
= U4) (4) (8)
2) dom HS ~[4¥) [3 9%)
at ey)
TT deenain ole = a) 4 |
anh JG) Oe Djand
res pestiwly Henge de ne
oo Cope Surkgeh Peg on EG)
. sya Hab 5 ted t
} ~ val at DMO Has 3c 407 |
aA Ris deine shen od raroon adi. |
am
~
sent
4
SF cing dhe, chen or 3s 0g x C=
Solon ye400
WY
doen W(AJEC CH 502) 5
AL by, ot waresy-<
at “wn Fees yes
ae sik poet 70 «
: ata: - GR 5) 78
Coes ese
seus? > 8 ge
WT
« dort ee AaeNe
Z| Fed the domoin= ob
OZ
y bow = VIE |
Soldo |
fort bivide Velue
- 4-nbo : 7
> tat x20
Et). yg 20D, Sy
Laan Ge Pa ie eee |
3: oS op se 7
ae a an or up) "s
Sar AHN Sate
be ae
: '
= (aray(e-3)20
< He 3 NERF
C4 aie 9 poy
1 ae 4, pq he
z le ae ue ‘ | |
or dom Hays (cee gage)Pind The domoin ob-
Vq- yp otx)= Via
Seldtont— |, \ Selullon2— ‘
Fo Hinile. YoLue : Fore. nite velas 1 |
1k 0 | ey Oe al 0 ~blw a i
F(a) 2020 | (= (530 : LS LTOn ir |
7 (31 B-9s0 | (ats) (20 <
= Me-3. Hog oo 4 PAs Ae . ,
Geesy aetayh ww" eset ane fey Chas ka) ata)!
qe or 4-4 Ath Ate Te 9 |: oa Aeag 2 9-35
Pepe as Fey ay . +16 = aah
a s domiPUQ= (3,3) ain bea) Ca FEPUES =)
BO see ) Vara ut
Soluiom = of ‘Magen a |
nae trom binite. Value. Lat bom Sine Vodue, a
ze -4>0 -%
> Ero
D (143) (1-8
Ne =; { i .
( : 2 2. \ | assy"
S)-1 (29 Wa, | ag Pages I
= as-4 4-4, be 4,) eee 200) Yave bar 4
= 16 aoe ete 8G 8
olen Ga) = (-cos-2y U3)
selubion ?—
or binite Value
AH S0
GJIE>O
(ery (s-0 8
fe
sacar
sides te rsd
2 ase
oA 3k
wy dom FCM) = 1-599)
shee)
deve)
pore
ray
“ee
Bew”
ae
asnug
jay
eb)"
accf
o\e
zedae a |
K-2c>0
at-(of >0
(ass) OT) 30
WET HET
t-GPass; 1 ee,
3G - as q-ar
=" = rls
oy dom Fas Ge s6) uLG0)
QS) b= ya Ha
Sebadlen = bog Gite value
A220 K-320 5
RL KE Ze 2g)
S dom bry) = (3, [3..02) i
RQ ‘2 71.39%, 4 |
Forge The geal Kot 2 or)
bunction Y= by
-8
FReomeson bor, binding, outreange “ob tumetion ¢ |
aoe
For the bunedion = bO4)
A} God A owe values obs of
i GLoriant valued or. with cerfest to 9
a Pilidabte
Sebelduli oy’ abl s "
fe the. Wanda 10 ellet a ee bre dm
iy Coches ot oxplicttly nae nik 1 bee
(ike!
le ae
7 A4> WL
yw = A-ayos "i
» pre = Fray)» -
- {
abs dae : |
= Vir ay |
‘ i
|
i
eerues
adding P37
~ Lin both side |
Braheteok
+ AY ESY = AS —
rane
Nay 4) = O92),
1 oeflikcsy astay “ i
2 ugh dan pRBd-AD fet ow fea |
_ Me Piasxzy ae Byte
8 Finke canny? SEAS) ae
1-ag sed
aged
os \
J remnyeta)= BAAFind the range oF the bunedion KO) = aH
gabon peas gna es |
a :
det, $= DANS
Ww
= WH AktF
may - ARB!)
BC 2) 28
weds ’
gat .
bon bine ceonge
y-2
SFB
rromy F(a)= R$ 24
YD) bey =v
p | Sebuten o=
BO) VIE yp pa
bbs Jav
3° aa)
L = Meh
= te
<4 x
bot aeihe tome oo
ion= BR |
bak or bene thet he-domaln atthe Swen reef
TT Monge H=TB 1) |
Phe) = Va ‘
Selubion—
VIS. 22
FOr A
OE for biakle domain’ |
bak AUF 0 |
ing Lin both Sid -3tvd nave! |
GP 0-2 '(0) $e |
2 NR ETE
“B+ pte t ean)
eh pedo fy doko e (290) | |
, |
7 KEYS ( i
mae = Se
TARR
eo
J rong bay,» (Ore)tO) afte
SHAT Fey Firlke deren
1-20
FUS-F 20
7 (AOU 20. = 8 A £4
ay he bash r
t2yo a-(0)
= id =i-d =-L-y
= 33 -3
209g
aL =
Sido BUA) = EA
Hea) = WEat
<= To
ELA) = Wo-3
Solution 1
AN-Z=0 .
Adding £2 in bet GEE.
q AnfrfsroF2
WFR re EF
dome GTS 9)
S> Vans
Seyort
dts
eet Gide i
So hes)
|
+
|
y Qr=3 i
Sis “a ovat SB |
|
|
Sea i"
as be we Ineg i aeb tha agen pe
bum etions 1g TM Sernain
9100), Corse”,
ACCOR Lin) gy >), e,
He!
Lat BRO BAe oot
PE f
en] toy =)
3 i Aedety oa
den BNE Tf EG
Fos VC roy lent (on
Loge Pg aa |
2 yoy sah | oven
SOF gia t yetty>*
3
O | bm» Be
Solution —
qe Fe
fon finite Tonge ‘ i
4 Baye
i 7 bee as-3 Ss
a t= BARS :
x |
AS ety-3, |
q
| A132 55
' wosbtiohy a both (de
asswtsort
a2 0
Gs z0
-4)20
Oy GE
i
1
|
|
4 -(-0) 9H ge (cL entesomy seg
Gelb a
ss ebm) = [23,3]
meng Hx) = £053]ay i 20
(sy G) 20
(a4) (sx) 20
get de
aeyee Gee gO)
att ari, adc bh
aoe
“i ¢
36 ~ 4
} Ream bX)
Ss Soares}
aaa
Yeoh SR
#35
oe E010) rang bl) = Rlebel
fo davinite: namge range 06 the. bunction P= =)
but ob emo ‘thet Ane doroin ob the Seat tench
bunesion is (0 400) $2 be ‘element ie crcéon ding tort”
Mapbra ok teal valu bunction
Eaualtly ok funetiing
Tole bunetion beryl. 9 ae bald Hobe eral
Den ere:
Leary
ay
We Viaayt
ay
S40
. » domain(s) ~ domain( y)
vi Co=\-Sormain (4) =c0zdomaingg)). sso
wy Fore any iin demain gra)= Jerry t "
“ELE FEN PBs Ae MAT BaF akey
Qik > B Bees a tay +3
} domain (Ledomenl|) es ey |
» Co- dowsing) = Co~dorrain (4). |
Koda gew |
7 ss J= 8%
bla Par _
FapbeMag otnts ih 4 bse bar]
al 4 took
BA) iQ) ee IG) 3.309
X35 Ke WL)? 2 (08) Sl@h
6
3 bE) =TE)SS
vy ©S Sw
enxi- GO) =
Solutent—
Orgy my Hey + glo:
= a (antd)
4 ipa see aan,
on |G). Sulled ok hay" ponctions,
a] ety be} 1X 7 R towns»
r | Addon ob two buncton i—
OO Rr rr
het, bs 92% > Boston » , |
The dolitjon ob--fev0 dem bumertion Ketuen as (ety 1
detingk by ( FEY = HA (1) ,2LE X '
| The, dlomain o- +9 is @yual 40 the nen
Mand You ize..Dr 4g = PEND). ae
Jon) Ergot:
eA bt) aa ee \ The, Quotiemt Ob d(x) and $l4):46 denoted by
Golatens—voilsiv) oa) ae deblend by, re te, Kepand 50) 4
(Bip = 600-4 7 DE. ven dy - {aout
5
= x + \ : wo.
san banca en BUR GO are
_Sibines Hn. ok ante =
Nets 90 9 Reithensy
Heian 4") elution s— eA),
The Sub-Heefion 8b--fd.tonctlen redunen a5), oy "yatta oe
cetined by" (gyn HFC, ERs | ae
| The dlorraje ob 6-9 ho +654 fo the eresectis y . (£2 Gnge ie) i
FEA) ond fa) gt a. | G | Muliplicadion of Seabee queatity ev
He Or Pa DiNDy 5 Liplication oF . 7 FADES Some,
BAe Hoya Metyeanads one \ | 7 elon suenieeneaast e KA)
Soutien (Eig 2 Fagg enim HOO,2 ame | Ka
UD = We. , poof Golltion = geene, a ote
WE = neon OO retie ot FOAY
YE Mulliphicocion ok jue Ganedion + — | ee 90") =V ETAL tind
Katy 19 WFR then, { P Hg '2 1b) +91) HX) -4¢
“Th walhittication of v0 SuncHn ref eken asl) Oa tae bY Ji
devine’ by (89) 4 = FU GKD- HEX i Nees ; be NTH MURK
“he denen ok 9 ic onial FOAL intention |) ED= HOR GY LO) BRE Re
FQ) ond SS) \ = Vian- view Vow
SST ve Drege DeND, ake Spe tO! ON * fra | = Gnd “He domein i= oe
9 7 FOO Net yy Ee | ee :
det, sR ZR. be debined aely by 5O%) = wa for. tinle demote Vows Cal
GLO = anna Grd ( Hos ot: Fond a So extrxo
Solution — ~ 28 i J= boy sols echt Virco... /
bag rig 45 sistan ai] et ‘
= AAAS 5 2 “EE isin) ws |
Sons) aihthy {| awd) ) |
Wa Geis) Coie a) - x ee ty ~ bt Vey-¢ 5 : |
2 ared)tt Qty 0) B® ‘ | fe Vey ef
= ane DRIES 9 a 7 he } ReveD | |
= ann 3 Va == |
TE CY = bdioge Cie) fae row tt Feperg ee ~ ACHES ve) Leis |
= F(a . : . fon dlont(oy a= § ya 4-4 a
toluhin 2 Gun ogee By Some reel bunction end hie grep he , 3
> Constant bunedton
7A, bumction k2 —R (DER) is ee i
uncon iW i ees & |
yA gre Si oooae 1s ‘odd Tenclion ont Cake 7
va Fonction bSR- py Rdatined by. 7 ren ome oe nae eet .
CMe ANKE PIG Called sly bumetion. | % & temetn UO) tall bo cst ton F
5.0116 dakmed by BRee LOO LT Our} | Fea)'= BEARER. 2
2 Fay bumetion dom f= gh | eK boa
7 aes « Fey ea4
ea Fat jog tia » “ KOD= FH to he lemon FO oe orp
Wage k a exe -
KED=53 ag | gree |
Seon | i
' x
(ar)
yy *
4 ye era |
; | |
| ; 4) the Tal go ent - onl
|The Te rans < Seager Sh a Gein which at tewelion BUD) tS Said “ped ont 5
Biinotall fades i a YeQ = 4
7 A temetion 7 Py i | ext RF 8H Ve ta
F(a) = Osta Fos HATE. a Tod we Sea Ps
amon —noretive Mbager. omd Aeris O2s>- SECA) EROS Ana oreelion F ollt {
ee Croat een annzx0 ie Called Polyframial 4 Gps ©: » ON oefion «|
om Sipe Pebynomial ob dignee 1 | B Yt : : yi |
Re HG = ard 4 art Smee | ot ok i {
j Qe) = Ss i os
Pf Radtoral bine signs — é
Fi
“The $ Foay= 20D an |
nae re gy
tonal “huncrion, i |
eRe bY. |
Sete ee ai
sto th oki
7A. bimction 1s tyemebide uth».
QoL Detined “the bonctlon odd or emen.
GQ) BA) = SxS Aare
BER) = SER )HAE-)
= -Bx8- a
> Gere) 1
FO) = —b ea
” Se ie Feeation ts odeh
i) bey = sat cayenton,
Selutione—
Fed = 308 cae)
= BxS-s tee
2 BLK)
SELAH torte tpretton Is-emen |
B> tey= eg
> The Qragh ak @ edd. bumetion ts Symmokeic with ear
plies Oe Fie oO Real “bumedion “hen the: binetions
“|Z bey - fa CAA VERS)
ee 4 Et (y= = &)
a Zag hy. eeld-Fla)= 0 fte)
eee
Ae ae
Be A we :
LT Toe Co
eee
pe OY
ow Fen et oow tk
iy toy Gay)
32k,
BEAT
=o), |
by EA)= bE to Ae temction is odel-
Sea a) fay CE |
En) og, ON
= boy (670+ Va
= boy eS cama]
w fey [EO WEL) y
AVS> Wey [Re |
>) (SRR
“6S ase
oly COrVard) ©
OR
seit eae mere all
rR Ris ned
lay = ef (m20 )
sh (L0)
non ie alto Krew as slide vale
bometion |
Joke domain “Re ond mone is By Go]
in bellu, Rue.
eK [alas
SL es~
“The Yrath of y= IKI Comivts ob Awd” caress o8thoun
NE
7] The rorya syn ic
‘Sum function S-
Te crywm bunedion on P dabined &3 5
~ fe eet
ma = { Te a oY
The domain oft RW Ao
(hot
“Exponential bunedion z— SgnH=0
An emponenfial Ieunction ts dabined by = ©
FQ) (are, a¢t) KER
don bays, pt
“raha, RePY ee. ie
i ko (o)| a
\\ |
. (4 ! (a |
| Watts abet lew ~ a
| WO) = aM, wyer i j EA? bo) = boyy (2%) i |
| P oket We X=0 | 4 Solution = Fog (1-2)
WY We aad of sal hoy | ype aK Lbs Lee
9 ace hoe a> of itkazy | enw ; | |
SN
; kone
a2 5 | pony Hy = 207 Inger bn : ag(egg ee
fe
Bre tos Reten ce
eas [0,02 J=0
Lre2g-£ f
(seo
igen teat
The. beveion shy ag
sel aoe eS
{ene hey |
ae peek RE R[nrine =]
Copeeomt TRY PPR | ag
Note Te Ce crak bined hig Wndkon
wegavey) Dy. conenen, binelbn and box Ht
w a odd burchiviy SINCE > B-JOOHREY 2|
abOD. set) phvaber)
wh Eee Ipunciion Can BO dora Mae na .
funciin ord on omen bumation = 4