関数の話

ؔ਺ͷ࿩ !ZVSPZPSP

ؔ਺ͱ͸

Z G Y ೖྗYʹରͯ͠ɺग़ྗZͷ஋Λܾఆ͢Δنଇ

" # $ % *OU 4USJOH
ࣸ૾ͭͷू߹ͷରԠؔ܎

JOU TUS JOH ू߹ܕͱߟ͑ΔͱɺܕͱܕͷରԠنଇ

ؔ਺͕ ϑΝʔετΫϥε

MFUGaY TIPX Y ม਺ʹʮؔ਺ʯΛ୅ೖ ଋറ Ͱ͖Δ )BTLFMM

G Y \YUP@T ^ ม਺ʹʮؔ਺ʯΛ୅ೖ ଋറ Ͱ͖Δ 3VCZ

DPOTUG Y \SFUVSOYUP4USJOH ^ ม਺ʹʮؔ਺ʯΛ୅ೖ ଋറ Ͱ͖Δ +BWB4DSJQU

ؔ਺ద༻

ධՁʮؔ਺ʯʹҾ਺Λ༩͑ͯ݁ՌΛಘΔ͜ͱ SFUVSOYUP4USJOH GJOUTUSJOH SFUVSO ʠʡ Ҿ਺Y

G H I 2Ͳͷؔ਺͔ΒʮධՁʯ͢Δ

G H I ਖ਼֨ධՁ಺ଆ͔Βɻී௨ͷݴޠ $+43VCZ+BWBʜ

G H I ඇਖ਼֨ධՁ֎ଆ͔Βɻ஗ԆධՁ )BTLFMM*0ʜ

ߴ֊ؔ਺

MFUH *OU4USJOH TUSJOH HBʠIFMMPʠ B ʮؔ਺ʯΛड͚औΔؔ਺ )BTLFMM

H GO \ʠIFMMPʠ GO ^ ʮؔ਺ʯΛड͚औΔؔ਺ 3VCZ

DPOTUH GO \SFUVSOʠIFMMPʠ GO ^ ʮؔ਺ʯΛड͚औΔؔ਺ +BWB4DSJQU

ʮؔ਺HʯͷҾ਺GOʹʮؔ਺GʯΛ༩͑Δ SFUVSOYUP4USJOH SFUVSOʠIFMMPʠ GO GJOUTUSJOH H JOUTUSJOH TUSJOH lIFMMPz
SFUVSO Ҿ਺GO

ड͚औͬͨʮؔ਺ʯGOΛ࢖ͬͯTUSJOHΛฦ͢ H JOUTUSJOH TUSJOH SFUVSOʠIFMMPʠ SFUVSOYUP4USJOH GOJOUTUSJOH lIFMMPz SFUVSO
Ҿ਺Y SFUVSOYUP4USJOH GOJOUTUSJOH

Ϋϩʔδϟ

MFUB MFUIaYY B पғͷʮ؀ڥʯΛऔΓࠐΜͩʮؔ਺ʯ )BTLFMM

B I Y \Y B^ पғͷʮ؀ڥʯΛऔΓࠐΜͩʮؔ਺ʯ 3VCZ

DPOTUB DPOTUI Y \SFUVSOY B^ पғͷʮ؀ڥʯΛऔΓࠐΜͩʮؔ਺ʯ +BWB4DSJQU

ʮؔ਺Iʯ͸ʮBʯͱ͍͏؀ڥΛ͍֮͑ͯΔ SFUVSOY IJOUJOU B B ؀ڥΛऔΓࠐΉ

ʮؔ਺ʯ͕ʮঢ়ଶʯΛ͍࣋ͬͯΔͱ΋ݴ͑Δ SFUVSOY IJOUJOU B SFUVSO

Ϋϩʔδϟͱ ߴ֊ؔ਺

MFUJ*OU *OU4USJOH JYaZTIPX Y Z Ҿ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛ࡞ͬͯฦ͢ʮؔ਺ʯ )BTLFMM Ϋϩʔδϟ

J Y \ Z \ Y Z UP@T^ ^
Ҿ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛ࡞ͬͯฦ͢ʮؔ਺ʯ 3VCZ Ϋϩʔδϟ

DPOTUJ Y \ SFUVSO Z \ SFUVSO Y Z UP4USJOH
^ ^ Ҿ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛ࡞ͬͯฦ͢ʮؔ਺ʯ +BWB4DSJQU Ϋϩʔδϟ

ड͚औͬͨҾ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛฦ͢ JJOUJOUTUSJOH SFUVSO JOUTUSJOH SFUVSO Y ؀ڥΛऔΓࠐΉ SFUVSO Z UP4USJOH
Y Y SFUVSO Z UP4USJOH Ҿ਺Y JOUTUSJOH

ؔ਺߹੒

GɾH Y G H Y ̎ͭͷؔ਺G HΛॱ൪ʹద༻͢Δ߹੒ؔ਺GɾHΛੜ੒͢Δ͜ͱ

MFUGaYY MFUHaYTIPXY MFUHGaYHGY ؔ਺HG͸Ҿ਺YΛGʹద༻ͨ݁͠ՌΛHʹద༻ͯ͠ฦ͢ )BTLFMM

G Y \Y ^ H Y \YUP@T^ HG Y \H
G Y ^ ؔ਺HG͸Ҿ਺YΛGʹద༻ͨ݁͠ՌΛHʹద༻ͯ͠ฦ͢ 3VCZ

DPOTUG Y \SFUVSOY ^ DPOTUH Y \SFUVSOYUP4USJOH ^ DPOTUHG Y
\SFUVSOH G Y ^ ؔ਺HG͸Ҿ਺YΛGʹద༻ͨ݁͠ՌΛHʹద༻ͯ͠ฦ͢ +BWB4DSJQU

߹੒ؔ਺HG͸ؔ਺GͱHΛ΋ͭʮΫϩʔδϟʯ SFUVSO HGJOUTUSJOH SFUVSOY GJOUJOU SFUVSOYUP4USJOH HJOUTUSJOH SFUVSOY
GJOUJOU SFUVSOYUP4USJOH HJOUTUSJOH

߹੒ؔ਺HG͸ؔ਺GͱHΛ΋ͭʮΫϩʔδϟʯ SFUVSO HGJOUTUSJOH SFUVSOY GJOUJOU SFUVSOYUP4USJOH HJOUTUSJOH SFUVSO
ʠʡ

ؔ਺߹੒͢Δؔ਺

DPNQPTF G H GɾH ̎ͭͷؔ਺G HΛҾ਺ʹ߹੒ؔ਺HGΛฦ͢ʮؔ਺ʯ

CD BC BD MFUHGHG )BTLFMMʹ͸૊ΈࠐΈͰ ؔ਺͕͋Δ
)BTLFMM

EFGDPNQPTF H G Y \H G Y ^
FOE 3VCZͰ͸Ҿ਺H G͔ΒMBNCEBΛੜ੒ͯ͠ฦ͢ 3VCZ

DMBTT1SPD EFGDPNQPTF G Y \TFMG G Y ^
FOE BMJBT@NFUIPE DPNQPTF FOE 1SPDΫϥεͷΠϯελϯεϝιουʹ͢Δͷ͕ࣗવ 3VCZ

G Y \Y ^ H Y \YUP@T^ HDPNQPTF G
HG 1SPDΫϥεͷΠϯελϯεϝιουʹ͢Δͷ͕ࣗવ 3VCZ

DPOTUDPNQPTF H G \ SFUVSO Y \SFUVSOH G Y ^
^ +BWB4DSJQUͰ͸Ҿ਺H G͔ΒGVODUJPOΛੜ੒ͯ͠ฦ͢ +BWB4DSJQU

'VODUJPOQSPUPUZQFDPNQPTF GVODUJPO G \ SFUVSO Y \SFUVSOUIJT G Y ^
^ 'VODUJPOʹDPNQPTFΛఆٛ͢Δํ͕ࣗવ +BWB4DSJQU

ؔ਺ W T ΦϒδΣΫτ

Φ ϒδΣ Ϋ τ ؔ ਺ ม ਺
ʹ୅ ೖͰ͖ Δ ม ਺ ʹ୅ ೖͰ͖ Δ

Φ ϒδΣ Ϋ τ ؔ ਺ ؔ ਺
ͷҾ ਺ʹ౉ ͤ Δ ؔ ਺ ͷҾ ਺ʹ౉ ͤ Δ

Φ ϒδΣ Ϋ τ ؔ ਺ ঢ় ଶ
Λ࣋ ͭ ϝϯό ʔ ม਺ ঢ় ଶ Λ࣋ ͭ Ϋϩʔ δ ϟ

Φ ϒδΣ Ϋ τ ؔ ਺ ؔ ਺
Ͱੜ ੒ͯ͠ ໭ Γ ஋ʹ Ͱ͖Δ ؔ ਺ Ͱੜ ੒ͯ͠ ໭ Γ ஋ʹ Ͱ͖Δ

Φ ϒδΣ Ϋ τ ؔ ਺ ૊ Έ
߹Θ ͤΔ͜ ͱ ͕Ͱ͖ Δ EF M FH BU F D PN Q PTJ UJP O ૊ Έ ߹Θ ͤΔ͜ ͱ ͕Ͱ͖ Δ ؔ ਺߹੒

ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

͔͜͜Β͸ ͕࣌ؒ༨ͬͨΒ

ؔ਺߹੒ͷ ॱং

I H G Y YΛGʹ౉ͨ݁͠ՌΛɺHʹ౉ͯ͠ಘͨ݁ՌΛɺIʹ౉͢

I H G Y ӈ͔Βࠨ΁ॲཧ͕ྲྀΕ͍ͯ͘

IɾHɾG Y ߹੒ؔ਺IɾHɾG͸ӈ͔Βద༻͞Ε͍ͯ͘

GH Y H G Y ߹੒ͷॱংΛٯʹͯ͠߹੒͢Δ Λߟ͑Δ GɾH Y
G H Y

Ͱ߹੒͢Δͱɺॲཧ͕ࠨ͔ΒӈʹྲྀΕΔ I H G Y GHI Y

JNQPSU$POUSPM$BUFHPSZ $BUFHPSZDBUDBUBCDBUCDDBUBD )BTLFMMͰ͸$POUSPM$BUFHPSZʹ ͕͋Δ )BTLFMM

DMBTT1SPD EFG H Y \H TFMG Y ^
FOE FOE 3VCZͰ͸1SPDͷԋࢉࢠ Λॻ͖׵͑Δ 3VCZ

'VODUJPOQSPUPUZQFBOE5IFO GVODUJPO G \ SFUVSO Y \SFUVSOG UIJT Y ^
^ 'VODUJPOʹBOE5IFOͳͲ࡞Ε͹Ͳ͏ʹ͔ +BWB4DSJQU

ؔ਺߹੒ͷॱংΛม͑Δͱ ࠨ͔Βӈ΁ॲཧͷྲྀΕΛ දݱ͢Δ͜ͱ͕Ͱ͖Δ ೔ຊਓͳͷͰࠨ͔ΒಡΈ͍ͨ

ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

ؔ਺ద༻͢Δ ؔ਺

ؔ਺ద༻΋ࠨ͔Βӈʹྲྀ͍ͨ͠ GHI Y

] ͸ؔ਺ద༻͢Δʮؔ਺ʯ Y]GG Y

Y͕ࠨ͔ΒӈʹG H IΛ௨ͬͯྲྀΕ͍ͯ͘ Y] GHI

] B BC C B]GGB தஔԋࢉࢠ ] ͸ӈลͷؔ਺ʹࠨลͷ஋Λ౉͢ )BTLFMM

DMBTT0CKFDU EFGʛʼ G G TFMG FOE FOE 3VCZͰ͸શ֯
ʛʼ ͰͲ͏ʹ͔ʜ 3VCZ

0CKFDUQSPUPUZQFSFWBQQMZ GVODUJPO G \SFUVSOG UIJT ^ 0CKFDUQSPUPUZQFԚછ +BWB4DSJQU

ؔ਺ద༻͢Δؔ਺Ͱ Ҿ਺͕ࠨ͔Βӈ΁ྲྀΕΔ༷Λ දݱ͢Δ͜ͱ͕Ͱ͖Δ ೔ຊਓͳͷͰࠨ͔ΒಡΈ͍ͨ

6OJYͷQJQF΍ &MJYJSͷ]ͳͲ Ͱ΋͋Μ·Γ͜͏͍͏ॻ͖ํ͠ͳ͍͚Ͳʜ

ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

ؔ਺߹੒ͷ จ຺

EFGDPNQPTF G Y \TFMG G Y ^ FOE
ؔ਺߹੒ͷ࣮૷Ͱʜ 3VCZ

EFG G Y \TFMG G Y
^ FOE G Y ͷ݁ՌΛഒʹͯ͠ΈΔ 3VCZ

G Y \Y ^ H Y \Y ^ I Y
\YUP@T^ IHG ʠʡ ߹੒͞Εͨؔ਺ͷؒͰҾ਺͕ഒʹ 3VCZ

߹੒͞Εͨؔ਺ͷؒͰҾ਺͕ഒʹ Y GJOUJOU Y HJOUJOU YUP@T IJOUTUSJOH
ʠʡ

EFG G Y \ SFTG Y SFTOJM
OJMTFMG SFT ^ FOE G Y ͕OJMͳΒ͹ɺ߹੒͞Εͨؔ਺Λݺ͹ͣʹऴྃ͢Δ 3VCZ

G Y \<OJM Y >TBNQMF^ H Y \<OJM Y >TBNQMF^
I Y \YUP@T^ IHG OJMPSʠʡ G HͷͲͪΒ͔͕OJMΛฦ͢ͱશମͷ݁Ռ͕OJMʹͳΔ 3VCZ

߹੒͞Εͨؔ਺ͷؒͰOJMνΣοΫ <OJM Y >TBNQMF GJOUJOU <OJM Y >TBNQMF HJOUJOU YUP@T
IJOUTUSJOH OJM OJM ʠʡ ऴྃ ऴྃ

ʮ݁Ռ͕ഒʹͳΔʯจ຺ ʮOJMͩͱதࢭ͢Δʯจ຺ ؔ਺߹੒ʹ༷ʑͳʮจ຺ʯΛ༩͑Δ͜ͱ͕Ͱ͖Δ

ʮঢ়ଶ͕มԽ͢Δʯจ຺ ʮೖग़ྗ͕ൃੜ͢Δʯจ຺ ʮඇಉظͰܭࢉ͢Δʯจ຺ ଞʹ΋͍Ζ͍Ζͳʮจ຺ʯ͕͋Δ

ؔ਺߹੒Λந৅Խ͢Δͱ ੜ੒͞ΕΔؔ਺ʹ༷ʑͳ ੑ࣭Λ༩͑Δ͜ͱ͕Ͱ͖Δ ΋͔ͯ͠͠Ϟφʜʜ ҧ͍·͢

ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

ྫ ؔ਺߹੒Ͱ 'J[[#V[[

DMBTT1SPD EFGPS G Y \ SFTTFMG Y
SFTOJM G Y SFT ^ FOE FOE ࣗ਎ʹYΛద༻ͨ݁͠Ռ͕OJMͳΒ͹ɺ߹੒͞Εͨؔ਺ΛݺͿ 3VCZ

GJ[[ Y \Y GJ[[OJM^ CV[[ Y \Y CV[[OJM^ GJ[[CV[[ Y
\ YY GJ[[CV[[OJM^ JE Y \YUP@T^ GJ[[ CV[[ GJ[[CV[[ͱ਺ࣈΛग़ྗ͢Δؔ਺Λ༻ҙ 3VCZ

NBQ GJ[[CV[[PSGJ[[PSCV[[PSJE ֤ؔ਺ΛPSͰ߹੒ͯ͠NBQ͢Ε͹GJ[[CV[[Ͱ͖Δ 3VCZ

ͦͷଞͷ τϐοΫ

wŋࢀরಁա wŋΧϦʔԽ wŋ෦෼ద༻ wŋ࠶ؼ wŋ৞ΈࠐΈؔ਺ wŋ୅਺σʔλߏ଄ ϞϊΠυͱ͔ ͜ͷ͋ͨΓͷ࿩΋ػձ͕͋ͬͨΒ

ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

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