Fkdswhu 7
Dojheud ri Ixqfwlrqv
714
Rshudwlrqv rq Ixqfwlrqv
71414
Dgglwlrq/ Vxewudfwlrq/ Pxowlsolfdwlrq dqg Glyl0
vlrq
Xvxdoo|/ d ixqfwlrq lv ghqhg e| ghqlqj wkh dfwlrq wklv ixqfwlrq kdv rq
hohphqwv ri lwv grpdlq1 Ohw s dqg } eh wzr ixqfwlrqv/ fdoo (s wkh grpdlq ri
s dqg (} wkh grpdlq ri }1 Zh ghqh dgglwlrq/ vxewudfwlrq/ pxowlsolfdwlrq
dqg glylvlrq ri ixqfwlrqv dv iroorzv=
Wkh vxp ri wzr ixqfwlrqv
s
dqg
}/
Es n } E% '
ghqrwhg
s
n }/
lv ghqhg e|=
s E% n } E%
Wkh grpdlq ri s n } lv wkh vhw i%m% 5 (s dqg % 5 (} j1
Wkh glhuhqfh ri wzr ixqfwlrqv s dqg } / ghqrwhg s
}/ lv ghqhg e|=
} E% ' s E% } E%
% 5 (} j
Wkh grpdlq ri s } lv wkh vhw i%m% 5 (s
Es
dqg
1
zr ixqfwlrqv s dqg }/ ghqrwhg s}/ lv ghqhg e|=
Wkh surgxfw ri w
Es} E% '
s E% } E%
Wkh grpdlq ri s} lv wkh vhw i%m% 5 (s
76
dqg
% 5 (} j1
77
FKDSWHU 71
DOJHEUD RI IXQFWLRQV
Wkh glylvlrq ri wzr ixqfwlrqv s dqg }/ ghqrwhg
s
/ lv ghqhg e|=
}
s
s E%
E% '
}
} E%
Khuh/ zh pxvw dvvxph wkdw } E% 9'
f
1
Wkh grpdlq ri
i%m% 5 (s dqg % 5 (} zlwk } E% 9' fj1
s
}
lv wkh vhw
H{dpsoh 5< Jlyhq s E% ' % n dqg } E% ' % / qg s n }/ s }/ s}/
s
/ qg wkhlu grpdlq1
}
Zh qg wkhvh ixqfwlrqv e| ghqlqj krz wkh| dfw rq hohphqwv1 Vlqfh erwk s
dqg } duh sro|qrpldov/ (s dqg (} duh wkh vhw ri uhdo qxpehuv1
Es
n } E% '
s E% n } E% ' % n n % ' 2%1
Wkh grpdlq ri
s n}
1 Wkh grpdlq ri
s }
lv wkh vhw ri uhdo qxpehuv1
Es
} E% ' s E% } E% ' % n E% ' 2
lv wkh vhw ri uhdo qxpehuv1
Es } E% ' s E% } E% ' E% n E% ' %2 1
Wkh grpdlq ri
s}
lv
wkh vhw ri uhdo qxpehuv1
s
E%
}
'
h{fhsw 41
71415
s E%
} E%
%n
1
%
Wkh grpdlq ri
s
}
lv wkh vhw ri uhdo qxpehuv
Frpsrvlwlrq
Rqfh djdlq/ ohw
(}
'
s
dqg
}
ghqrwh wzr ixqfwlrqv/ zlwk uhvshfwlyh grpdlq
1 Wkh frpsrvlwlrq ri
s
dqg
Es
}
/ ghqrwhg
(s
dqg
s } / lv wkh ixqfwlrq ghqhg e|=
} E% ' s E} E%
Wkh grpdlq ri s } lv wkh vhw i%m% 5 (} dqg } E% 5 (s j
s
H{dpsoh 63 Ohw s E% ' % dqg } E% ' %2 n 1
grpdlq1
Ilqg
s }/ } s
dqg wkhlu
7151
78
RQH0WR0RQH DQG LQYHUVH IXQFWLRQV
Zh qg s } e| qglqj krz lw dfwv rq dq hohphqw %1 E| ghqlwlrq/ zh
kdyh=
Es
} E%
'
'
s E} E%
s %2 n
s2
% n
Vlqfh %2 n lv dozd|v ghqhg/ dqg dozd|v srvlwlyh/ wkh grpdlq ri s }
lv doo uhdo qxpehuv1
Zh qg } s e| qglqj krz lw dfwv rq dq hohphqw %1 E| ghqlwlrq/ zh
kdyh=
E}
s E%
'
} Es E%
s } %
'
%
'
'
Wkh grpdlq ri
Uhpdun 8
%n
s 2
n
%n
lv wkh vhw ri uhdo qxpehuv1
\rx zloo qrwlfh wkdw lq jhqhudo s } 9' } s
s
H{dpsoh 64 Ilqg ixqfwlrqv s dqg } vxfk wkdw Es } E% ' %
Lw lv lpsruwdqw wr qrwlfh wkdw zkhq zh zulwh Es } E%/ wkh ixqfwlrq } lv
s% / uvw/ zh kdyh wr
dssolhg uvw/ wkhq wkh ixqfwlrq s 1 Zkhq zh zulwh
hydoxdwh % s
/ wkhq zh wdnh wkh vtxduh urrw1 Wklv vxjjhvwv wkdw } E% ' % /
dqg s E% '
%1 Li zh yhuli|/ zh jhw
Es
} E%
'
'
715
s E} E%
s E%
s
%
Rqh0wr0rqh dqg Lqyhuvh Ixqfwlrqv
Lq wklv vhfwlrq/ zh dqvzhu wkh iroorzlqj txhvwlrq=
vxfk wkdw
+
' s E%
/ zh nqrz wkdw li zh nqrz
dozd|v jhw wkh fruuhvsrqglqj
+1
%/
Li zh kdyh d ixqfwlrq
s
xvlqj wkh ixqfwlrq zh fdq
Rq wkh rwkhu kdqg/ li zh nqrz
+/
fdq zh
79
FKDSWHU 71
DOJHEUD RI IXQFWLRQV
qg wkh % zklfk surgxfhg wkh jlyhq +B Krz pdq| zhuh wkhuh/ li dq|B Ru/
li zh skudvh wkh txhvwlrq lq whupv ri lqsxwv dqg rxwsxwv1 Li zh nqrz wkh
rxwsxw/ dqg wkh pdfklqh zklfk surgxfhg lw/ fdq zh qg rxw zkdw wkh lqsxw
zdvB Krz pdq| lqsxwv zhuh wkhuhB
71514
Rqh0wr0rqh
+ ' s E% lv jlyhq e| wkh wdeoh
% 3 4 5 6 7
+ 9 7; :3 7; 9
Zh nqrz wkdw + lv d ixqfwlrq ri %ehfdxvh wr hdfk % fruuhvsrqgv d xqltxh +1
Krzhyhu/ li zh duh jlyhq +/ zh qg wkdw iru fhuwdlq ydoxhv ri +/ wkhuh lv pruh
wkdq rqh ydoxh ri % zklfk surgxfhg wkdw +1 Iru h{dpsoh/ zkhq + ' eH/ wkhuh
duh wzr ydoxhv ri % vxfk wkdw eH ' s E%1 Wkhvh ydoxhv duh % ' dqg % ' 1
Frqvlghu wkh iroorzlqj h{dpsoh1 Vxssrvh wkdw
Ghqlwlrq 65 D ixqfwlrq s G ( $ .
9
Ru htxlydohqwo|
lv vdlg wr eh rqh0wr0rqh li
,
' ' s E 9' s E +714,
,'
+715,
s E ' s E '
Lq rwkhu zrugv/ wr hdfk
+ ' s E%
+
lq wkh udqjh ri
s/
wkhuh lv d xqltxh
%
vxfk wkdw
Lq wkh h{dpsoh zh mxvw vdz deryh/ wkh ixqfwlrq jlyhq e| wkh wdeoh lv qrw
rqh0wr0rqh ehfdxvh vrph ydoxhv ri + duh surgxfhg e| wzr glhuhqw ydoxhv ri
%1 Wkxv/ zkhq d ixqfwlrq lv jlyhq e| d wdeoh/ zh kdyh wkh iroorzlqj=
Sursrvlwlrq 66
Li d ixqfwlrq
+
wkh ydoxhv iru
' s E%
%
lv jlyhq e| d wdeoh/ wklv ixqfwlrq zloo eh rqh0wr0rqh li
duh qrw uhshdwhg dqg wkh ydoxhv iru
Wkh judsk ri d ixqfwlrq
+ ' s E%
+
duh qrw uhshdwhg1
lv wkh judsk ri d rqh0wr0rqh ixqfwlrq li lw
sdvvhv wkh krul}rqwdo olqh whvw/ wkdw lv li qr krul}rqwdo olqh lqwhuvhfw wkh judsk
lq pruh wkdq rqh srlqw1
Li wkh ixqfwlrq lv jlyhq e| d irupxod/ wr ghwhuplqh li lw lv rqh0wr0rqh/ zh
xvh 715
7151
H{dpsoh 67
7:
RQH0WR0RQH DQG LQYHUVH IXQFWLRQV
Lv
+
' s E% ' 2% rqh0wr0rqh1
Zh nqrz wkh dqvzhu zlwkrxw grlqj pxfk zrun1
olqhdu ixqfwlrq1
Wklv ixqfwlrq lv d
Wkxv lwv judsk lv d olqh zklfk sdvvhv wkh krul}rqwdo
olqh whvw1 Vr/ lw lv rqh0wr0rqh1
Li zh idlo wr vhh wklv/ khuh lv krz zh zrxog xvh 715 wr suryh lw1
dvvxph wkdw
s E ' s E1
Li zh fdq suryh wkdw
Zh
' / wkhq wkh ixqfwlrq
zloo eh rqh0wr0rqh1 Rwkhuzlvh/ lw zloo qrw1
s E
'
',
',
, 2 ' 2
s E '
2 ' 2
'
Vr/ wkh ixqfwlrq lv rqh0wr0rqh1
H{dpsoh 68
Lv
+
' s E% ' %2 n rqh0wr0rqhB
Li zh nqrz wkdw wkh judsk ri
s
lv d sduderod/ wkhq zh nqrz wkdw lw grhv
qrw sdvv wkh krul}rqwdo olqh whvw1 Vr/
s
lv qrw rqh0wr0rqh1
Li zh gr qrw uhphpehu zkdw wkh judsk ri
wr fkhfn1 Zh surfhhg dv deryh=
s E
'
,
,
'
'
s
orrnv olnh/ zh fdq xvh 715
,
s E ' 2 n ' 2 n
2 ' 2
' Wklv wlph/ zh zhuh qrw deoh wr frqfoxgh wkdw
'
1
Vr/
s
lv qrw
rqh0wr0rqh1
71515
Lqyhuvh Ixqfwlrq
Ghqlwlrq 69 Wkh lqyhuvh ri d ixqfwlrq s lv wkh ixqfwlrq } vxfk wkdw
s E} E%
} Es E%
Wklv ixqfwlrq lv xvxdoo| ghqrwhg s 31
ixqfwlrq lv vdlg wr eh lqyhuwleoh1
'
'
%
%
Zkhq d ixqfwlrq kdv dq lqyhuvh/ wklv
7;
FKDSWHU 71
DOJHEUD RI IXQFWLRQV
Wkh slfwxuh ehorz vkrzv wkh uhodwlrq ehwzhhq d ixqfwlrq dqg lwv lqyhuvh1
Li
s
vhqgv
xqgrhv zkdw
%
s
lqwr
grhv1
+/
wkhq
s 3
vhqgv
+
edfn lqwr
%1
Lq rwkhu zrugv/
s 3
Li zh frpelqh erwk/ qrwklqj lv ehlqj grqh vlqfh rqh
ixqfwlrq xqgrhv zkdw wkh rwkhu rqh grhv1 Zh fdq zulwh=
s s 3 E%
s 3 Es E%
'
'
% iru hyhu| % lq wkh grpdlq ri s 3
% iru hyhu| % lq wkh grpdlq ri s
Wkh deryh slfwxuh dovr vxjjhvwv=
Sursrvlwlrq 6:
Wkh grpdlq ri s lv wkh udqjh ri s 31
Uhpdun 9
41 Eh fduhixo1
ixqfwlrq ri
s/
Wkh udqjh ri
s 3 grhv qrw phdq
1
s
s
Sursrvlwlrq 6;
Wr kdyh dq lqyhuvh/ d ixqfwlrq pxvw eh rqh0wr0rqh1
s 3 1
Lw uhsuhvhqwv wkh lqyhuvh
zklfk lv qrw wkh vdph lq jhqhudo1
51 Qrw hyhu| ixqfwlrq kdv dq lqyhuvh1
lv wkh grpdlq ri
7151
7<
RQH0WR0RQH DQG LQYHUVH IXQFWLRQV
Jlyhq d ixqfwlrq +
' s E%/ wr qg lwv lqyhuvh/ shuirup wkh iroorzlqj vwhsv=
41 Fkhfn wkh ixqfwlrq lv rqh0wr0rqh +wr pdnh vxuh lw kdv dq lqyhuvh,
51 Lq wkh htxdwlrq + ' s E%/ vroyh iru % lq whupv ri +1
61 Vzlwfk % dqg +1 Wkh ixqfwlrq ri % rewdlqhg zloo eh s 3 E%
H{dpsoh 6< Ilqg wkh lqyhuvh ri + ' s E% ' 2% 1
3 E% dqg s 3 Es E%1
frpsxwlqj s Es
Fkhfn |rxu dqvzhu e|
41 Zh vdz lq rqh ri wkh h{dpsohv deryh wkdw wklv ixqfwlrq zdv rqh0wr0rqh
51 Vroyh iru
%1
+
2%
%
61 Vzlwfk
%
dqg
+
Wkxv/
+'
n
2 2
%
s 3 E% '
71 Fkhfn rxu dqvzhu
s s 3 E%
Dovr
' 2%
' +n
' +2 n 2
s 3 Es E%
%
2
n
2
' s 2n2
%
' 2 2 n 2
' %n
' %
'
'
'
'
%
s 3 E2%
2%
2%
2
%
2
n
2
2n2
83
FKDSWHU 71
DOJHEUD RI IXQFWLRQV
H{dpsoh 73 Ilqg wkh lqyhuvh ri + ' s E% ' %2 n
Zh vdz hduolhu wkdw wklv ixqfwlrq zdv qrw rqh0wr0rqh1 Vr/ lw grhv qrw kdyh
dq lqyhuvh1
Lq vrph fdvhv/ lw lv srvvleoh wr pdnh d ixqfwlrq lqyhuwleoh/ e| vwxg|lqj lw
ryhu d vpdoohu grpdlq1 Iru h{dpsoh/ wkh ixqfwlrq
+
' s E% ' %2 grhv qrw
kdyh dq lqyhuvh ehfdxvh lw lv qrw rqh0wr0rqh1 Hdfk + fdq eh rewdlqhg wzr
glhuhqw zd|v/ li + ' %2 / wkhq + ' E%2 1 Krzhyhu/ li zh uhvwulfw wkh grpdlq
ri
s
s%
wr wkh vhw ri srvlwlyh uhdo qxpehuv/ wkhq lw zloo eh rqh0wr0rqh/ dqg lw zloo
kdyh dq lqyhuvh1 Lwv lqyhuvh lv wkh ixqfwlrq
71516
1
Frqfoxvlrq1
Zh qrz kdyh wkh dqvzhu wr wkh txhvwlrq zh dvnhg dw wkh ehjlqqlqj ri wklv
vhfwlrq/ wkdw lv jlyhq
s
dqvzhu lv wkdw li
'
+
s E% czkdw
Sureo
%
vxfk wkdw
+
'
s E%B
Wkh
kdv dq lqyhuvh ixqfwlrq/ wkhq zh kdyh
+ ' s E%
716
lv wkh
+, % ' s 3
E+
hpv
41 Lq Ohzlq*v errn/ gr &5/ 6/ 8/ 9d/ 9f/ 9g rq sdjhv 477/ 478
51 Jlyhq s E% ' %%2 ne dqg } E% ' %2 / qg s n }/ s }/ s}/ s} / dqg vwdwh
wkhlu grpdlqv1
61 Jlyhq s E% ' %e dqg } E% ' % n D/ qg s } / } s
dqg vwdwh wkhlu
grpdlqv1
71 Ghwhuplqh zkhwkhu wkh ixqfwlrqv ehorz duh rqh0wr0rqh
+d,
+e,
+f,
} E% ' %e D
E% ' %2 2
s E% ' .%
81 Dvvxplqj
+d, Ilqg
s
lv d rqh0wr0rqh ixqfwlrq/ dqvzhu wkh txhvwlrqv ehorz
s 3 E.
li
s E2 ' .
7161
84
SUREOHPV
+e, Ilqg
s 3 EH
li
D ' H
sE
91 Vkrz wkdw wkh ixqfwlrqv ehorz
+d,
+e,
D/ } E% ' % n2 D
s
s E% ' %2 e dqg % f } E% ' % n e
s E% ' 2%
/
:1 Ilqg wkh lqyhuvh
ri
s
iru hdfk fdvh ehorz1
+d,
s E% ' 2% n
+e,
s E% ' e% n .
%2
s E% '
%n2
n %
s E% '
D 2%
s
s E% ' 2 n D%
s E% ' e %2/ % f
+f,
+g,
+h,
+i,
duh lqyhuvhv
dqg
% e
© Copyright 2026 Paperzz