✶
❋✉♥çõ❡s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧
❊♠ ♠✉✐t❛s s✐t✉❛çõ❡s✱ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❝❡rt❛ ❣r❛♥❞❡③❛ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞♦ ✈❛❧♦r ❞❡ ✉♠❛ s❡❣✉♥❞❛ ❣r❛♥❞❡③❛✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❞❡♠❛♥❞❛ ❞❡ ✉♠ ♣r♦❞✉t♦ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞♦ s❡✉ ♣r❡ç♦❀ ♦ ♥í✈❡❧ ❞❡ ♣♦❧✉✐çã♦ ❞♦ ❛r ❡♠ ✉♠❛ ❝✐❞❛❞❡ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞♦ ♥ú♠❡r♦ ❞❡ ✈❡í❝✉❧♦s ♥❛s r✉❛s❀ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❣❛rr❛❢❛ ❞❡ ✈✐♥❤♦ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞♦ ❛♥♦ ❡♠ q✉❡ ♦ ✈✐♥❤♦ ❢♦✐ ❢❛❜r✐❝❛❞♦❀ ♦ ❝✉st♦ ❞❡ ✉♠❛ ❝❛♠♣❛♥❤❛ ❞❡ ✈❛❝✐♥❛çã♦ ♣♦❞❡ ❞❡♣❡♥❞❡r ❞♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ✈❛❝✐♥❛❞❛s✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❡ss❛ ❞❡♣❡♥❞ê♥❝✐❛ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞❛ ❛tr❛✈és ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❡ ♣♦r ✐ss♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ❢✉♥çõ❡s ♥ã♦ é r❡str✐t❛ ❛♣❡♥❛s ❛♦s ✐♥t❡r❡ss❡s ❞❛ ▼❛t❡♠át✐❝❛✱ ♠❛s ❝♦❧♦❝❛❞❛ ❡♠ ♣rát✐❝❛ ❡♠ ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ❝♦♠♦ ❛ ❋ís✐❝❛✱ ❛ ◗✉í♠✐❝❛✱ ❛ ❇✐♦❧♦❣✐❛ ❡ ❛s ❈✐ê♥❝✐❛s ❊❝♦♥ô♠✐❝❛s✱ ❡♥tr❡ ✈ár✐❛s ♦✉tr❛s✳
❆s ❢✉♥çõ❡s t❛♠❜é♠ ♥ã♦ ❛♣❛r❡❝❡♠ s♦♠❡♥t❡ ♥♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦✳ ❆ s✐t✉❛çã♦ ❛ s❡❣✉✐r ✐❧✉str❛ ❡ss❡ ❢❛t♦✳
❯♠❛ ♦♣❡r❛❞♦r❛ ❞❡ t❡❧❡❢♦♥✐❛ ❝❡❧✉❧❛r ❢❡③ ✉♠❛ ♣r♦♠♦çã♦ q✉❡ ♦❢❡r❡❝❡ três ♦♣çõ❡s ❞❡ ♣❧❛♥♦ ❞❡ ♣❛❣❛♠❡♥t♦ ❛♦s s❡✉s ❝❧✐❡♥t❡s✳ ❖❜s❡r✈❡ ♦ ❛♥✉♥❝✐♦ ❛❜❛✐①♦ ❡ r❡s♣♦♥❞❛ ❛s ♣❡r❣✉♥t❛s✳
✶✳ ❈✐t❡ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦s ♣❧❛♥♦s ♠❡♥s❛✐s ❆ ❡ ❇ q✉❡ ♥ã♦ ❛♣❛r❡❝❡ ♥♦ ♣❧❛♥♦ ❞❡ r❡❝❛r❣❛ ❈❀ ✷✳ ◗✉❛❧ s❡r✐❛ ♦ ♣❧❛♥♦ ♠❛✐s ❡❝♦♥ô♠✐❝♦ ♣❛r❛ ✉♠❛ ✉t✐❧✐③❛çã♦ ❞❡ ✷✵ ♠✐♥✉t♦s ♣♦r ♠ês❄
✸✳ ❉❡t❡r♠✐♥❡ ♦ ❝✉st♦ ❞❡ ✸✵ ♠✐♥✉t♦s ❞❡ ❧✐❣❛çõ❡s ♣♦r ♠ês ❡♠ ❝❛❞❛ ♣❧❛♥♦ ♦❢❡r❡❝✐❞♦ ♣❡❧❛ ♦♣❡r❛❞♦r❛✳ ❈♦♠♣❛r❡ ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s❀
✹✳ ◗✉❛❧ s❡r✐❛ ♦ ♣❧❛♥♦ ♠❛✐s ❡❝♦♥ô♠✐❝♦ ♣❛r❛ ✉t✐❧✐③❛r ✻✵ ♠✐♥✉t♦s ❞❡ ❧✐❣❛çõ❡s ♣♦r ♠ês❄
✺✳ P❡♥s❛♥❞♦ ♥❛s r❡s♣♦st❛s ❛♥t❡r✐♦r❡s✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦ ❝✉st♦ ❞❡ ❝❛❞❛ ♣❧❛♥♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦ ✉t✐❧✐③❛❞♦❄ ❏✉st✐✜q✉❡ s✉❛ r❡s♣♦st❛❀
✻✳ ◗✉❡ ❢❛t♦r❡s ❞❡✈❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞♦ ♣❧❛♥♦ ❝♦♠ ♦ ♠❡❧❤♦r ❝✉st♦✲❜❡♥❡❢í❝✐♦❄
❆ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ❢✉♥çã♦
✷
❊①❡♠♣❧♦ ✶✳ ❆ ❈♦♠♣❛♥❤✐❛ ❊s♣ír✐t♦ ❙❛♥t❡♥s❡ ❞❡ ❙❛♥❡❛♠❡♥t♦ ✭❈❡s❛♥✮ ❝♦❜r❛ ❛s s❡❣✉✐♥t❡s t❛r✐❢❛s ♣❛r❛ ♦ ❢♦r♥❡❝✐♠❡♥t♦ ❞❡ á❣✉❛ r❡s✐❞❡♥❝✐❛❧ ♣❛❞rã♦✿
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ t❛❜❡❧❛✱ ❛ t❛r✐❢❛ ❛ s❡r ♣❛❣❛ ❞❡♣❡♥❞❡ ❞❛ ❢❛✐①❛ ❞❡ ❝♦♥s✉♠♦ ❞❡ á❣✉❛✱ ♦✉ s❡❥❛✱ ❛ t❛r✐❢❛ ❡stá ❡♠ ❢✉♥çã♦ ❞❛ ❢❛✐①❛ ❞❡ ❝♦♥s✉♠♦✳
❊①❡♠♣❧♦ ✷✳ ❖ ❝♦♠♣r✐♠❡♥t♦ C ❞❡ ✉♠ ❝ír❝✉❧♦ ❞❡♣❡♥❞❡ ❞❡ s❡✉ r❛✐♦ r✳ ❉✐③✲s❡ q✉❡ C é ✉♠❛ ❢✉♥çã♦ ❞❡ r✳ ❆ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ q✉❡ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ C é C= 2πr✳ ❊ss❛ é ❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❢❛③ ❝❛❞❛ ✈❛❧♦r ♣♦s✐t✐✈♦ ❞❡ r ❝♦rr❡s♣♦♥❞❡r ❛ ✉♠ ú♥✐❝♦ ✈❛❧♦r ❞❡C✳
❊①❡♠♣❧♦ ✸✳ ❖s ❜✐ó❧♦❣♦s ❞❡s❝♦❜r✐r❛♠ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s❛♥❣✉❡ ❡♠ ✉♠❛ ❛rtér✐❛ é ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ s❛♥❣✉❡ ❡ ♦ ❡✐①♦ ❝❡♥tr❛❧ ❞❛ ❛rtér✐❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❧❡✐ ❞❡ P♦✐s❡✉✐❧❧❡✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s❛♥❣✉❡ q✉❡ ❡stá ❛ r cm ❞♦ ❡✐①♦ ❝❡♥tr❛❧ ❞❛ ❛rtér✐❛ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ S(r) =C(R2−r2)✱ ♦♥❞❡ C= 1,76×105 cm❡R= 1,2×10−2 cm✳
❊①❡♠♣❧♦ ✹✳ ❆ t❡♠♣❡r❛t✉r❛ T r❡❣✐str❛❞❛ ❡♠ ◦C ♣❡❧♦ ■♥st✐t✉t♦ ◆❛❝✐♦♥❛❧ ❞❡ ▼❡t❡♦r♦❧♦❣✐❛ ✭■♥♠❡t✮ ❞✉r❛♥t❡ ✉♠ ❞✐❛ ❞❡ ♣r✐♠❛✈❡r❛ é ✉♠❛ ❢✉♥çã♦ ❞♦ t❡♠♣♦t ❞❛❞♦ ❡♠ ❤♦r❛s✳
❊♠❜♦r❛ ♥ã♦ ❤❛❥❛ ✉♠❛ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ s✐♠♣❧❡s q✉❡ r❡❧❛❝✐♦♥❡ ❛s ❞✉❛s ❣r❛♥❞❡③❛s✱ ❡ss❛ s✐t✉❛çã♦ ❞❡s❝r❡✈❡ ✉♠❛ ❧❡✐ s❡❣✉♥❞♦ ❛ q✉❛❧ ♣❛r❛ ❝❛❞❛ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦t❤á ✉♠❛ ú♥✐❝❛ t❡♠♣❡r❛t✉r❛T r❡❣✐str❛❞❛✳ ◆❡ss❛ ❢✉♥çã♦✱ ❛ t❡♠♣❡r❛t✉r❛ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦ ❡✱ ♣♦r ✐ss♦✱ é ❝❤❛♠❛❞❛ ❞❡ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡✳ ❏á ♦ t❡♠♣♦✱ ❝♦♠♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ♥❛❞❛✱ é ❝❤❛♠❛❞♦ ❞❡ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳
❚❛❜❡❧❛s✱ ❢ór♠✉❧❛s ❡ ❣rá✜❝♦s sã♦ ❛s ❢♦r♠❛s ♠❛✐s ❝♦♠✉♥s ✉t✐❧✐③❛❞❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠❛ ❢✉♥çã♦✳
❉❡✜♥✐çã♦ ❋♦r♠❛❧
❉❛❞❛s ❞✉❛s ✈❛r✐á✈❡✐s x❡y✱ ❡♠ q✉❡ xé ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ ❡ y ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❞❡x✱ s❡ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡ xé ♣♦ssí✈❡❧ ❛ss♦❝✐❛r ✉♠ ú♥✐❝♦ ✈❛❧♦r ❞❡y✱ ❡♥tã♦y ❡stá ❡♠ ❢✉♥çã♦ ❞❡x✳
❉❡✜♥✐çã♦ ✶✳ ❯♠❛ ❢✉♥çã♦ f é ✉♠❛ ❧❡✐ q✉❡ ❢❛③ ❝❛❞❛ ❡❧❡♠❡♥t♦x ❞❡ ✉♠ ❝♦♥❥✉♥t♦A ❝♦rr❡s♣♦♥❞❡r ❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ y ❞❡ ✉♠ ❝♦♥❥✉♥t♦ B✳
❊♠ ❣❡r❛❧ ✉s❛♠♦s ❛ ♥♦t❛çã♦
f :A −→ B x 7−→ y
✸
❉♦♠í♥✐♦✱ ❈♦♥tr❛✲❞♦♠í♥✐♦ ❡ ■♠❛❣❡♠
❉❛❞❛ ✉♠❛ ❢✉♥çã♦f ❞❡A❡♠B✱ ♦ ❝♦♥❥✉♥t♦Aé ❞❡♥♦♠✐♥❛❞♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦f✱ ❡ ♦ ❝♦♥❥✉♥t♦B ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ss❛ ❢✉♥çã♦✳ ❖ ❞♦♠í♥✐♦ é ❞❡♥♦t❛❞♦ ♣♦rD(f)♦✉ s✐♠♣❧❡s♠❡♥t❡D❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ♣♦rCD(f)♦✉CD✳ ❙❡ ✉♠ ❡❧❡♠❡♥t♦x∈A ❡st✐✈❡r ❛ss♦❝✐❛❞♦ ❛ ✉♠ ❡❧❡♠❡♥t♦y∈B✱ ❞✐③❡♠♦s q✉❡y é ❛ ✐♠❛❣❡♠ ❞❡x✭✐♥❞✐❝❛✲s❡y=f(x)❡ ❧ê✲s❡ ✏② é ✐❣✉❛❧ ❛f ❞❡x✑✮✳ ❖ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞❛s ❛s ✐♠❛❣❡♥s ❞❡ Dé ❞❡♥♦♠✐♥❛❞♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❞❡f✳ ❆ ♥♦t❛çã♦ ✉t✐❧✐③❛❞❛ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ éIm(f)♦✉Im✳
❊①❡♠♣❧♦ ✺✳ ❙❡❥❛f é ✉♠❛ ❢✉♥çã♦ ❞❡ N❡♠N ✭✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ❞♦♠í♥✐♦ ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ sã♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✮
❞❡✜♥✐❞❛ ♣♦r y=x+ 2✳ ❊♥tã♦ t❡♠♦s q✉❡✿
• ❆ ✐♠❛❣❡♠ ❞❡1 ❛tr❛✈és ❞❡ f é 3✱ ♦✉ s❡❥❛✱f(1) = 1 + 2 = 3❀
• ❆ ✐♠❛❣❡♠ ❞❡2 ❛tr❛✈és ❞❡ f é 4✱ ♦✉ s❡❥❛✱f(2) = 2 + 2 = 4✳
❉❡ ♠♦❞♦ ❣❡r❛❧✱ ❛ ✐♠❛❣❡♠ ❞❡x❛tr❛✈és ❞❡ f é x+ 2✱ ♦✉ s❡❥❛✿ f(x) =x+ 2✳
✹
❊st✉❞♦ ❞♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧
❯♠❛ ✈❡③ q✉❡ ♥ã♦ ❡stá ❞❛❞♦ ♦ ❞♦♠í♥✐♦✱ ♥ã♦ ❡stá ❝❛r❛❝t❡r✐③❛❞❛ ❢✉♥çã♦ ❛❧❣✉♠❛✳ ❊①✐st❡✱ ♣♦ré♠✱ ✉♠❛ ❝♦♥✈❡♥çã♦ ❣❡r❛❧ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡✿ s❡♠♣r❡ q✉❡ ♥ã♦ ❢♦r ❡①♣❧✐❝✐t❛❞♦ ♦ ❞♦♠í♥✐♦ D ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✜❝❛ s✉❜❡♥t❡♥❞✐❞♦ q✉❡ ❡❧❡ ❡stá ❞❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ R✱ ❞❡✈❡♥❞♦ s❡r ❡①❝❧✉í❞♦s ❛♣❡♥❛s ♦s ✈❛❧♦r❡s ♣❛r❛ ♦s q✉❛✐s ❛s ♦♣❡r❛çõ❡s ✐♥❞✐❝❛❞❛s ♣❡❧❛ ❧❡✐ ❞❡
❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♥ã♦ ❢❛③❡♠ s❡♥t✐❞♦✳ ❊①❡♠♣❧♦s ✶✳ • ❆ ❢✉♥çã♦ f(x) = 1
x t❡♠ ❞♦♠í♥✐♦ D = R∗ ♦✉ D = R\ {0}✱ ♣♦✐s✱ ♣❛r❛ t♦❞♦ x ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ♦
♥ú♠❡r♦ 1
x é ✉♠ ♥ú♠❡r♦ r❡❛❧✳
✺
P❧❛♥♦ ❈❛rt❡s✐❛♥♦ ❡ ●rá✜❝♦ ❞❡ ✉♠❛ ❋✉♥çã♦
P❧❛♥♦ ❈❛rt❡s✐❛♥♦
❖ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ é ❢♦r♠❛❞♦ ♣♦r ❞✉❛s r❡t❛s r❡❛✐s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❡♥tr❡ s✐ ❡ q✉❡ s❡ ❝r✉③❛♠ ♥♦ ♣♦♥t♦ ③❡r♦✳ ❊ss❡ ♣♦♥t♦ é ❞❡♥♦♠✐♥❛❞♦ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ❡ é ❢r❡q✉❡♥t❡♠❡♥t❡ ❞❡♥♦t❛❞♦ ♣♦rO✳ ❈❛❞❛ r❡t❛ r❡♣r❡s❡♥t❛ ✉♠ ❡✐①♦ ❡ sã♦ ♥♦♠❡❛❞♦s ♣♦rOx❡Oy✳ ❙♦❜r❡♣♦♥❞♦ ✉♠ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ❡ ✉♠ ♣❧❛♥♦✱ ♦❜té♠✲s❡ ✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❝✉❥❛ ♣r✐♠❡✐r❛ ✈❛♥t❛❣❡♠ é ❛ss♦❝✐❛r ❛ ❝❛❞❛ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ✉♠ ♣❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❆ss✐♠✱ ✉♠ ♣♦♥t♦P ❞♦ ♣❧❛♥♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ♣❛r ♦r❞❡♥❛❞♦(m, n)❝♦♠m❡nr❡❛✐s✳ ❖ ❡✐①♦ ❤♦r✐③♦♥t❛❧Oxé ❝❤❛♠❛❞♦ ❞❡ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❡ ♦ ❡✐①♦ ✈❡rt✐❝❛❧Oy✱ ❞❡ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s✳ ❊ss❡s ❡✐①♦s ❞✐✈✐❞❡♠ ♦ ♣❧❛♥♦ ❡♠ q✉❛tr♦ r❡❣✐õ❡s ❝❤❛♠❛❞❛s q✉❛❞r❛♥t❡s✳
❘❡♣r❡s❡♥t❛çõ❡s ●rá✜❝❛s ❞❡ ❋✉♥çõ❡s
❊①✐st❡♠ ✈ár✐❛s ❢♦r♠❛s ✈✐s✉❛✐s ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠❛ ❢✉♥çã♦✱ ❛❧❣✉♠❛s ❞❛s q✉❛✐s ❥á ✉t✐❧✐③❛♠♦s ♥♦s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s✳ ❉✉❛s ❞❡❧❛s sã♦ ♠❛✐s ❛❞❡q✉❛❞❛s q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❢✉♥çõ❡s ❡♥tr❡ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✳ ❆ t❡r❝❡✐r❛ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉s❛❞❛ ♣❛r❛ ❢✉♥çõ❡s ❝♦♠ ❞♦♠í♥✐♦ ✐❣✉❛❧ ❛R✳
✶✳ ❚❛❜❡❧❛s✿ ❡♠ ✉♠❛ ❝♦❧✉♥❛ r❡♣r❡s❡♥t❛♠♦s ♦s ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ ❡ ❡♠ ♦✉tr❛ ❝♦❧✉♥❛ r❡♣r❡s❡♥t❛♠♦s ♦s ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ✈❛❧♦r❡s ❛♣r❡s❡♥t❛❞♦s ♥❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛✳
❊①❡♠♣❧♦ ✻✳ ❙❡❥❛
f :{1,2,4,7} −→ R
x 7−→ x2−1
❆ ❢✉♥çã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♥❛ t❛❜❡❧❛ ❛❜❛✐①♦✿
x f(x) 1 f(1) = 12−1 = 0
2 f(2) = 42−1 = 3
4 f(4) = 42−1 = 15
7 f(7) = 72−1 = 48
✷✳ ❉✐❛❣r❛♠❛✿ r❡♣r❡s❡♥t❛♠♦s ♦ ❉♦♠í♥✐♦ ❡ ♦ ❈♦♥tr❛✲❞♦♠í♥✐♦ ❛tr❛✈és ❞❡ r❡❣✐õ❡s ❞♦ ♣❧❛♥♦ ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ❝♦♥❥✉♥t♦s ❝♦♠♦ ♣♦♥t♦s ❞❡♥tr♦ ❞❡ss❛s r❡❣✐õ❡s✳ ❆ ❢✉♥çã♦ é r❡♣r❡s❡♥t❛❞❛ ♣♦r s❡t❛s q✉❡ ❧✐❣❛♠ ❝❛❞❛ ♣♦♥t♦ ❞♦ ❞♦♠í♥✐♦ à s✉❛ ✐♠❛❣❡♠ ♥♦ ❝♦♥tr❛✲❞♦♠í♥✐♦✳
❊①❡♠♣❧♦ ✼✳ ❆ ❢✉♥çã♦
f :{1,2,4,7} −→ R
x 7−→ x2−1
♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❛tr❛✈és ❞♦ ❞✐❛❣r❛♠❛✿
✻
❊①❡♠♣❧♦ ✽✳ ❆ ❢✉♥çã♦
f :{1,2,4,7} −→ R
x 7−→ x2−1
♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❛tr❛✈és ❞♦ ❣rá✜❝♦✿
❖❜s❡r✈❡ q✉❡ ♣❛r❛ ✉♠ ❝❡rt♦ ✈❛❧♦r ❞❡ x♥♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✱ ♥♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦f ♣♦❞❡ ❤❛✈❡r ♥♦ ♠á①✐♠♦ ✉♠ ♣♦♥t♦ ❝♦♠ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ✐❣✉❛❧ ❛x✳ ■ss♦ ❛❝♦♥t❡❝❡ ♣♦rq✉❡ ♦s ♣♦♥t♦s ❞♦ ❣rá✜❝♦ sã♦ ❞❛ ❢♦r♠❛(x, f(x))❡ ❝♦♠♦f é ✉♠❛ ❢✉♥çã♦✱ só ❡①✐st❡ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ♣❛r❛f(x)✳ ❊ss❛ ♦❜s❡r✈❛çã♦ ♣❡r♠✐t❡ ❝r✐❛r ✉♠ t❡st❡ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠❛
❝✉r✈❛ ♥♦ ♣❧❛♥♦xyé ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✿ ♦ t❡st❡ ❞❛ r❡t❛ ✈❡rt✐❝❛❧✳
❆♦ tr❛ç❛r ✉♠❛ r❡t❛ ✈❡rt✐❝❛❧ ♣♦r ♣♦♥t♦s ❞♦ ❞♦♠í♥✐♦✱ ❡st❛ ❞❡✈❡ ✐♥t❡r❝❡♣t❛r ♦ ❣rá✜❝♦ ♥✉♠ ú♥✐❝♦ ♣♦♥t♦✱ ❥á q✉❡ ♣❛r❛ ❝❛❞❛x❞♦ ❞♦♠í♥✐♦ ❞❡✈❡ ❡①✐st✐r ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♠ ú♥✐❝♦y=f(x)♥♦ ❝♦♥tr❛❞♦♠í♥✐♦✳
❙❡ ❡st❛ r❡t❛ ✈❡rt✐❝❛❧ ❝♦rt❛r ♦ ❣rá✜❝♦ ❡♠ ♠❛✐s ❞❡ ✉♠ ♣♦♥t♦ ❡♥tã♦ ❡st❡ ❣rá✜❝♦ ♥ã♦ r❡♣r❡♥t❛ ✉♠❛ ❢✉♥çã♦✳
✼
❋✐❣✉r❛ ✷✿ ❈✉r✈❛ ♥♦ ♣❧❛♥♦xy q✉❡ ♥ã♦ r❡♣r❡s❡♥t❛ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❖❜s❡r✈❡ q✉❡ ❛s r❡t❛s ✈❡rt✐❝❛✐s ✐♥t❡r❝❡♣t❛♠ ❛ ❝✉r✈❛ ❡♠ ♠❛✐s ❞❡ ✉♠ ♣♦♥t♦✳
❊①❡♠♣❧♦ ✾✳ ❯♠❛ ❢✉♥çã♦
f :{−2,−1,0,1,2} −→ [0,4]
x 7−→ x2
✽
❚✐♣♦s ❞❡ ❋✉♥çõ❡s
❆s ❢✉♥çõ❡s ♣♦❞❡♠ ❡①✐❜✐r ♣r♦♣r✐❡❞❛❞❡s út❡✐s q✉❡ ❛s ❞❡st❛❝❛♠ ❞❛s ❞❡♠❛✐s ❢✉♥çõ❡s✳ P♦r ✐ss♦✱ ❛❧❣✉♠❛s ❞❡❧❛s r❡❝❡❜❡♠ ♥♦♠❡s ❡s♣❡❝✐❛✐s q✉❡ ✐♥❞✐❝❛♠ ❛s ❝❛r❛❝t❡ríst✐❝❛s q✉❡ ❡❧❛s ♣♦ss✉❡♠✳
• ❋✉♥çã♦ s♦❜r❡❥❡t♦r❛✿ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ é s♦❜r❡❥❡t♦r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ s❡✉ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❢♦r ✐❣✉❛❧ ❛♦
❝♦♥tr❛❞♦♠í♥✐♦✱ ✐st♦ é✱ s❡ Im=B✳ ❊♠ t❡r♠♦s ❞❛ r❡♣r❡s❡♥t❛çã♦ ❛tr❛✈és ❞❡ ❞✐❛❣r❛♠❛s✱ ✐ss♦ q✉❡r ❞✐③❡r q✉❡ ♥ã♦ ♣♦❞❡
s♦❜r❛r ❡❧❡♠❡♥t♦s ♥♦ ❝♦♥❥✉♥t♦B s❡♠ r❡❝❡❜❡r ✢❡❝❤❛s✳
• ❋✉♥çã♦ ■♥❥❡t♦r❛✿ ❆ ❢✉♥çã♦ é ✐♥❥❡t♦r❛ s❡ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞♦ ❞♦♠í♥✐♦ t✐✈❡r❡♠ ✐♠❛❣❡♥s ❞✐st✐♥t❛s✱ ♦✉ s❡❥❛✱ ❞♦✐s
❡❧❡♠❡♥t♦s ♥ã♦ ♣♦❞❡♠ t❡r ❛ ♠❡s♠❛ ✐♠❛❣❡♠✳ P♦rt❛♥t♦ ♥ã♦ ♣♦❞❡ ❤❛✈❡r ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ♥♦ ❝♦♥❥✉♥t♦ B q✉❡ r❡❝❡❜❛ ❞✉❛s ✢❡❝❤❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❢✉♥çã♦f :R→R❞❡✜♥✐❞❛ ♣♦rf(x) = 3xé ✐♥❥❡t♦r❛ ♣♦✐s s❡x16=x2❡♥tã♦3x16= 3x2✱ ❡
♣♦rt❛♥t♦ f(x1)6=f(x2)✳
• ❋✉♥çã♦ ❇✐❥❡t♦r❛✿ ❯♠❛ ❢✉♥çã♦ é ❜✐❥❡t♦r❛ q✉❛♥❞♦ ❡❧❛ é s♦❜r❡❥❡t♦r❛ ❡ ✐♥❥❡t♦r❛ ❛♦ ♠❡s♠♦ t❡♠♣♦✳ P♦r ❡①❡♠♣❧♦✱ ❛
❢✉♥çã♦f :R6=R❞❡✜♥✐❞❛ ♣♦ry = 3xé ✐♥❥❡t♦r❛✱ ❝♦♠♦ ✈✐♠♦s ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❊❧❛ t❛♠❜é♠ é s♦❜r❡❥❡t♦r❛✱ ♣♦✐s Im=B =R✳ ▲♦❣♦✱ ❡st❛ ❢✉♥çã♦ é ❜✐❥❡t♦r❛✳ ❏á ❛ ❢✉♥çã♦f :N→N❞❡✜♥✐❞❛ ♣♦r y=x+ 5♥ã♦ é s♦❜r❡❥❡t♦r❛✱ ♣♦✐s
Im={5,6,7,8, ...} ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦CD=N✱ ❡ ♣♦rt❛♥t♦ ♥ã♦ ♣♦❞❡ s❡r s♦❜r❡❥❡t♦r❛✳ ❊❧❛ é ✐♥❥❡t♦r❛✱ ❥á q✉❡ ✈❛❧♦r❡s
✾
• ❋✉♥çã♦ ♣❛r✿ ❉✐③❡♠♦s q✉❡f é ♣❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f(x) =f(−x)♣❛r❛ t♦❞♦x♥♦ ❞♦♠í♥✐♦ ❞❡ f✳ ❖✉ s❡❥❛✿ ✈❛❧♦r❡s s✐♠étr✐❝♦s ❞❡✈❡♠ ♣♦ss✉✐r ❛ ♠❡s♠❛ ✐♠❛❣❡♠✳
❊①❡♠♣❧♦ ✶✵✳ ❆ ❢✉♥çã♦ f : R→R ❞❡✜♥✐❞❛ ♣♦r f(x) =x2 é ✉♠❛ ❢✉♥çã♦ ♣❛r✱ ♣♦✐s f(−x) = (−x)2 =x2 =f(x)✳
P♦❞❡♠♦s ♥♦t❛r ❛ ♣❛r✐❞❛❞❡ ❞❡ss❛ ❢✉♥çã♦ ♦❜s❡r✈❛♥❞♦ ♦ s❡✉ ❣rá✜❝♦✿
◆♦t❛♠♦s✱ ♥♦ ❣rá✜❝♦✱ q✉❡ ❡①✐st❡ ✉♠❛ s✐♠❡tr✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ✈❡rt✐❝❛❧✳ ❊❧❡♠❡♥t♦s s✐♠étr✐❝♦s tê♠ ❛ ♠❡s♠❛ ✐♠❛❣❡♠✳ ❖s ❡❧❡♠❡♥t♦s2 ❡−2✱ ♣♦r ❡①❡♠♣❧♦✱ sã♦ s✐♠étr✐❝♦s ❡ ♣♦ss✉❡♠ ❛ ✐♠❛❣❡♠ 4✳
• ❋✉♥çã♦ í♠♣❛r✿ ❉✐③❡♠♦s q✉❡ f é ♣❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f(−x) = −f(x)♣❛r❛ t♦❞♦ x ♥♦ ❞♦♠í♥✐♦ ❞❡ f✳ ❖✉ s❡❥❛✿ ✈❛❧♦r❡s s✐♠étr✐❝♦s ❞❡✈❡♠ ♣♦ss✉✐r ✐♠❛❣❡♥s s✐♠étr✐❝❛s✳
❊①❡♠♣❧♦ ✶✶✳ ❆ ❢✉♥çã♦f :R→R❞❡✜♥✐❞❛ ♣♦rf(x) =x3 é ✉♠❛ ❢✉♥çã♦ ♣❛r✱ ♣♦✐s f(−x) = (−x)3=−x3=−f(x)✳
P♦❞❡♠♦s ♥♦t❛r q✉❡ ❡ss❛ ❢✉♥çã♦ é í♠♣❛r ♦❜s❡r✈❛♥❞♦ ♦ s❡✉ ❣rá✜❝♦✿
◆♦t❛♠♦s✱ ♥♦ ❣rá✜❝♦✱ q✉❡ ❡①✐st❡ ✉♠❛ s✐♠❡tr✐❛ ❡♠ r❡❧❛çã♦ ❛ ♦r✐❣❡♠ ✵✳ ❊❧❡♠❡♥t♦s s✐♠étr✐❝♦s tê♠ ✐♠❛❣❡♥s s✐♠étr✐❝❛s✳ ❖s ❡❧❡♠❡♥t♦s1 ❡−1✱ ♣♦r ❡①❡♠♣❧♦✱ sã♦ s✐♠étr✐❝♦s ❡ ♣♦ss✉❡♠ ✐♠❛❣❡♥s1❡ −1 ✭q✉❡ t❛♠❜é♠ sã♦ s✐♠étr✐❝❛s✮✳
❯♠❛ ❢✉♥çã♦ q✉❡ ♥ã♦ é ♣❛r ♥❡♠ í♠♣❛r é ❝❤❛♠❛❞❛ ❢✉♥çã♦ s❡♠ ♣❛r✐❞❛❞❡✳
• ❋✉♥çã♦ ❈r❡s❝❡♥t❡✿ ❉✐③❡♠♦s q✉❡ f é ❝r❡s❝❡♥t❡ s❡ ♣❛r❛ q✉❛✐sq✉❡r x1< x2 ♣❡rt❡♥❝❡♥t❡s ❛♦ ❞♦♠í♥✐♦ ❞❡ f t✐✈❡r♠♦s
f(x1) < f(x2)✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❢✉♥çã♦ f : R → R ❞❡✜♥✐❞❛ ♣♦r f(x) = x+ 1 é ❝r❡s❝❡♥t❡ ❡♠ R✱ ♣♦✐sx1 < x2 ⇒
✶✵
• ❋✉♥çã♦ ❉❡❝r❡s❝❡♥t❡✿ ❉✐③❡♠♦s q✉❡ f é ❞❡❝r❡s❝❡♥t❡ s❡ ♣❛r❛ q✉❛✐sq✉❡r x1 < x2 ♣❡rt❡♥❝❡♥t❡s ❛♦ ❞♦♠í♥✐♦ ❞❡ f
t✐✈❡r♠♦s f(x1) > f(x2)✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❢✉♥çã♦ f : R → R ❞❡✜♥✐❞❛ ♣♦r f(x) = −x é ❞❡❝r❡s❝❡♥t❡ ❡♠ R✱ ♣♦✐s
x1< x2⇒ −x1>−x2⇒f(x1)> f(x2)✳ ❖✉ s❡❥❛✿ q✉❛♥❞♦ ♦s ✈❛❧♦r❡s ❞♦ ❞♦♠í♥✐♦ ❝r❡s❝❡♠✱ s✉❛s ✐♠❛❣❡♥s ❞❡❝r❡s❝❡♠✳
❈♦♠♣♦s✐çã♦ ❞❡ ❋✉♥çõ❡s
❯♠❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ é ❝r✐❛❞❛ ❛♣❧✐❝❛♥❞♦ ✉♠❛ ❢✉♥çã♦ ❛♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ♦✉tr❛ ❢✉♥çã♦✱ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❈♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡✈❡ ♣♦ss✉✐r ✉♠ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦ ❜❡♠ ❞❡✜♥✐❞♦s ❡ ❡st❛♠♦s ❢❛❧❛♥❞♦ ❞❡ ❛♣❧✐❝❛r ❢✉♥çõ❡s ♠❛✐s ❞❡ ✉♠ ✈❡③✱ ❞❡✈❡♠♦s s❡r ♣r❡❝✐s♦s ❝♦♠ r❡❧❛çã♦ ❛ ❝♦♠♦ ❡st❛♠♦s ❛♣❧✐❝❛♥❞♦ ❡st❛s ❢✉♥çõ❡s✳
❉❡✜♥✐çã♦ ✷✳ ❉❛❞❛s ❛s ❢✉♥çõ❡sf :A→B ❡g:B →C ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡ g ❡f✱g◦f é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ g◦f :A −→ C
x 7−→ g(f(x)
❊①❡♠♣❧♦ ✶✷✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ❝♦♥❥✉♥t♦s A = {1,2}, B ={3,4} ❡ C ={5,6}✱ ❡ ❛s ❢✉♥çõ❡s f : A → B ❞❡✜♥✐❞❛ ♣♦r f(x) =x+ 1✱ ❡g:B →C❞❡✜♥✐❞❛ ♣♦rg(y) =y+ 1✳ ❆ ❝♦♠♣♦st❛g◦f é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦g◦f(x) =g(f(x)) =f(x) + 1 = (x+ 1) + 1 =x+ 2 ❡ ❡stá r❡♣r❡s❡♥t❛❞❛ ♥♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦✳
❊①❡♠♣❧♦ ✶✸✳ ❈♦♥s✐❞❡r❡♠♦s ♦s ❝♦♥❥✉♥t♦sA={−2,−1,0,1,2}, B={−2,1,4,7,10} ❡C={0,3,15,48,99}✱ ❡ ❛s ❢✉♥çõ❡s
f :A→B ❞❡✜♥✐❞❛ ♣♦r f(x) = 3x+ 4✱ ❡g:B→C ❞❡✜♥✐❞❛ ♣♦r g(y) =y2−1✳ ❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s
✶✶
x f(x) g◦f(x) =g(f(x))
−2 −2 3
−1 1 0
0 4 15
1 7 48
2 10 99
✶✷
❊①❡r❝í❝✐♦s
✶✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦y2=xé ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦y=f(x)❄
✷✳ ❖ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣❧❛♥♦ (x, y)∈R2:x2+y2= 1é ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦y=f(x)❄
✸✳ ❊♥❝♦♥tr❡ ♦ ❞♦♠í♥✐♦ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s✿
✭❛✮ 1 x2+4❀
✭❜✮ p
(x−1)(x+ 2)❀
✭❝✮ √3−2x−x2❀
✭❞✮ q3x−4
x+2✳
✹✳ ❙❡f(x) = 4x−3✱ ♠♦str❡ q✉❡f(2x) = 2f(x) + 3✳
✺✳ ◗✉❛✐s ♦s ❞♦♠í♥✐♦s ❞❡f(x) =x−18 ❡g(x) =x3❄ ❉❡t❡r♠✐♥❡ ♦ ❞♦♠í♥✐♦ ❞❡ h(x) =f(g(x))✳
✻✳ ❙❡f(x) = 1−x✱ ♠♦str❡ q✉❡f(f(x)) =x✳ ✼✳ ❙❡f(x) =ax+b
x−a ✱ ♠♦str❡ q✉❡f(f(x)) =x✳
✽✳ ❙❡ f(x) = ax✱ ♠♦str❡ q✉❡ f(x) +f(1−x) =f(1)✳ ❱❡r✐✜q✉❡ t❛♠❜é♠ q✉❡ f(x1+x2) = f(x1) +f(x2)✱ ♣❛r❛ t♦❞♦s
x1, x2∈R✳
✾✳ ❯♠❛ ❢✉♥çã♦ ❢ é ❧✐♥❡❛r s❡ f(u+v) =f(u) +f(v)❡f(au) =af(u)♣❛r❛ t♦❞♦ u, v∈R✳ ◗✉❛✐s ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s
sã♦ ❧✐♥❡❛r❡s❄
✭❛✮ f(x) = 2x❀
✭❜✮ f(x) = 2x+ 3❀
✭❝✮ f(x) =|x|❀
✭❞✮ f(x) =x2✳
✶✵✳ ▼♦str❡ q✉❡ s❡f ❧✐♥❡❛r ❡♥tã♦f(0) = 0✳
✶✶✳ ❯♠ ♣♦♥t♦ ✜①♦ ❞❡ ✉♠❛ ❢✉♥çã♦f é ✉♠ ♥ú♠❡r♦at❛❧ q✉❡f(a) =a✳ ❊♥❝♦♥tr❡ ♦s ♣♦♥t♦ ✜①♦s✱ s❡ ❡①✐st✐r❡♠✱ ❞❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s
✭❛✮ f(x) =x❀ ✭❜✮ f(x) = 2x−5❀
✭❝✮ f(x) =x2❀
✭❞✮ f(x) =x−1 x+1✳
✶✷✳ ❈❛r❛❝t❡r✐③❡ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s ❝♦♠♦ s♦❜r❡❥❡t♦r❛✱ ✐♥❥❡t♦r❛✱ ❜✐❥❡t♦r❛✱ ♦✉ ♥❡♥❤✉♠❛ ❞❡❧❛s✿
✭❛✮ f :R→R, f(x) = 3x+ 5❀
✭❜✮ f :R→R, f(x) = 2x+ 1❀
✭❝✮ f :R→R, f(x) =x2−9❀
✭❞✮ f :R→R, f(x) =x2+ 4❀
✭❡✮ f :A→A, f(x) =x2+ 4, A={x:x≥4}❀
✭❢✮ f :{x∈R:x≥0} →R, f(x) =5x32✳
✶✸✳ ❉❡t❡r♠✐♥❡ s❡ ❛s s❡❣✉✐♥t❡s ❢✉♥çõ❡s sã♦ ♣❛r❡s✱ í♠♣❛r❡s ♦✉ ♥❡♥❤✉♠❛ ❞❡❧❛s✿
✭❛✮ f(x) = 2x5+ 3x2❀ ✭❜✮ f(x) = 3−x2+ 2x4❀
✭❝✮ f(x) = 1−x❀
✭❞✮ f(x) =x+x3❀ ✭❡✮ f(x) =|x|❀
✭❢✮ f(x) =x(x3−x)❀
✭❣✮ f(x) =x3
+x x2+1✳
✶✹✳ ▼♦str❡ q✉❡ ♣❛r❛ t♦❞❛ ❢✉♥çã♦f :R→R❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ♣❛r g❡ ✉♠❛ ❢✉♥çã♦ í♠♣❛r ht❛❧ q✉❡f(x) =g(x) +h(x)✳
✶✺✳ ❙✉♣♦♥❤❛f(x)✉♠❛ ❢✉♥çã♦ í♠♣❛r ❡g(x)✉♠❛ ❢✉♥çã♦ ♣❛r✳
✭❛✮ P♦❞❡♠♦s ❢❛❧❛r ❛❧❣♦ s♦❜r❡ ❛ ♣❛r✐❞❛❞❡ ❞❡Q(x) =f(x)g(x)❡P(x) = f(x)g(x)❄
✶✸
❆♣❧✐❝❛çõ❡s
✶✳ ❉✉r❛♥t❡ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ✈❛❝✐♥❛çã♦✱ ♦s r❡s♣♦♥sá✈❡✐s ❝❛❧❝✉❧❛r❛♠ q✉❡ ♦ ❝✉st♦ ♣❛r❛ ✈❛❝✐♥❛r x% ❞❛ ♣♦♣✉❧❛çã♦ é ❞❛❞♦
♣♦r
C(x) = 150x
200−x ♠✐❧❤õ❡s ❞❡ r❡❛✐s. ✭❛✮ ◗✉❛❧ é ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦C❄
✭❜✮ P❛r❛ q✉❡ ✈❛❧♦r❡s ❞❡x❛ ❢✉♥çã♦C(x)t❡♠ s✐❣♥✐✜❝❛❞♦ ♥❡ss❡ ❝♦♥t❡①t♦❄
✭❝✮ ◗✉❛❧ é ♦ ❝✉st♦ ♣❛r❛ ✈❛❝✐♥❛r50%❞❛ ♣♦♣✉❧❛çã♦❄
✭❞✮ ◗✉❛❧ é ♦ ❝✉st♦ ♣❛r❛ ✈❛❝✐♥❛r ♦s50% r❡st❛♥t❡s ❞❛ ♣♦♣✉❧❛çã♦❄
✭❡✮ ◗✉❛❧ ♣♦r❝❡♥t❛❣❡♠ ❞❛ ♣♦♣✉❧❛çã♦ t❡rá s✐❞♦ ✈❛❝✐♥❛❞❛ ❛♣ós s❡r❡♠ ❣❛st♦s37,5 ♠✐❧❤õ❡s ❞❡ r❡❛✐s❄
✷✳ ❊st✐♠❛✲s❡ q✉❡ ❞❛q✉✐ ❛t ❛♥♦s ✉♠ ❝❡rt♦ ❜❛✐rr♦ t❡rá ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡P(t) = 20− 6
t+1 ♠✐❧ ❤❛❜✐t❛♥t❡s✳
✭❛✮ ◗✉❛❧ s❡rá ❛ ♣♦♣✉❧❛çã♦ ❞♦ ❜❛✐rr♦ ❛♣ós9 ❛♥♦s❄
✭❜✮ ◗✉❛❧ s❡rá ♦ ❛✉♠❡♥t♦ ❞❛ ♣♦♣✉❧❛çã♦ ❞✉r❛♥t❡ ♦9◦ ❛♥♦❄
✭❝✮ ❖ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠P(t)♣❛r❛ ❣r❛♥❞❡s ✈❛❧♦r❡s ❞❡t❄ ■♥t❡r♣r❡t❡ ♦ r❡s✉❧t❛❞♦✳
✸✳ P❛r❛ ❡st✉❞❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ ♦s ❛♥✐♠❛✐s ❛♣r❡♥❞❡♠✱ ✉♠ ❡st✉❞❛♥t❡ ❞❡ ♣s✐❝♦❧♦❣✐❛ r❡❛❧✐③♦✉ ✉♠ ❡①♣❡r✐♠❡♥t♦♥♦ q✉❛❧ ✉♠ r❛t♦ t❡✈❡ q✉❡ ♣❡r❝♦rr❡r ✈ár✐❛s ✈❡③❡s ✉♠ ❧❛❜✐r✐♥t♦✳ ❙✉♣♦♥❤❛ q✉❡ ♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ r❛t♦ ❡♥❝♦♥tr❛r ❛ s❛í❞❛ ♥❛n✲és✐♠❛ t❡♥t❛t✐✈❛ s❡❥❛ ❞❛❞♦ ♣♦r
T(n) = 3 +12
n. ✭❛✮ ◗✉❛❧ é ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦T❄
✭❜✮ P❛r❛ q✉❡ ✈❛❧♦r❡s ❞❡n❛ ❢✉♥çã♦T(n)t❡♠ s✐❣♥✐✜❝❛❞♦ ♥❡ss❡ ❝♦♥t❡①t♦❄
✭❝✮ ❆ ❢✉♥çã♦T é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡❄
✭❞✮ ◗✉❛♥t♦ t❡♠♣♦ ♦ r❛t♦ ❧❡✈♦✉ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ s❛í❞❛ ♥❛ q✉❛rt❛ t❡♥t❛t✐✈❛❄
✭❡✮ ❆ ♣❛rt✐r ❞❡ q✉❛❧ t❡♥t❛t✐✈❛ ♦ r❛t♦ ❝♦♥s❡❣✉❡ ❡♥❝♦♥tr❛r ❛ s❛í❞❛ ❝♦♠ ♠❡♥♦s ❞❡4♠✐♥✉t♦s❄
✭❢✮ ❖ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠T(n)♣❛r❛ ❣r❛♥❞❡s ✈❛❧♦r❡s ❞❡n❄ ■♥t❡r♣r❡t❡ ♦ r❡s✉❧t❛❞♦✳ ❖ r❛t♦ ❝♦♥s❡❣✉❡ ❡♥❝♦♥tr❛r ❛ s❛í❞❛ ❝♦♠ ♠❡♥♦s ❞❡3 ♠✐♥✉t♦s❄
✹✳ ❯t✐❧✐③❛♥❞♦ ❛s ✐♥❢♦r♠❛çõ❡s ❛♣r❡s❡♥t❛❞❛s ♥♦ ❡①❡♠♣❧♦ ✸✱ r❡s♣♦♥❞❛✿ ✭❛✮ ❉❡t❡r♠✐♥❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s❛♥❣✉❡ ♥♦ ❡✐①♦ ❝❡♥tr❛❧ ❞❛ ❛rtér✐❛✳
✭❜✮ ❉❡t❡r♠✐♥❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ s❛♥❣✉❡ ❛ ♠❡✐♦ ❝❛♠✐♥❤♦ ❡♥tr❡ ♦ ❡✐①♦ ❝❡♥tr❛❧ ❡ ❛ ♣❛r❡❞❡ ❞❛ ❛rtér✐❛✳ ✭❝✮ ❊♠ q✉❡ ♣♦♥t♦ ❞❛ ❛rtér✐❛ ❛ ✈❡❧♦❝✐❞❛❞❡ é ♠❛✐♦r✿ ♣ró①✐♠♦ ❛♦ ❡✐①♦ ❝❡♥tr❛❧ ♦✉ ❞✐st❛♥t❡ ❞♦ ❡✐①♦ ❝❡♥tr❛❧❄ ✺✳ ❖❜s❡r✈❛çõ❡s ♠♦str❛♠ q✉❡ ❡♠ ✉♠❛ ✐❧❤❛ ❞❡ A Km2♦ ♥ú♠❡r♦ ❞❡ ❡s♣é❝✐❡s ❞❡ ❛♥✐♠❛✐s é ❞❛❞♦ ♣♦r s(A) = 3√3
A.
✭❛✮ ◗✉❛♥t❛s ❡s♣é❝✐❡s ❡①✐st❡♠✱ ❡♠ ♠é❞✐❛✱ ❡♠ ✉♠❛ ✐❧❤❛ ❞❡8Km2❄
✭❜✮ ❆ ❢✉♥çã♦sé ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡ ❞❡ A❄ ❖ q✉❡ ✐ss♦ s✐❣♥✐✜❝❛❄ ✭❝✮ ❙❡s1=s(A0)❡s2=s(2A0)q✉❛❧ é ❛ r❡❧❛çã♦ ❡♥tr❡s1 ❡s2❄
✭❞✮ ◗✉❛❧ ❞❡✈❡ s❡r ❛ ár❡❛ ❞❛ ✐❧❤❛ ♣❛r❛ q✉❡ ❡❧❛ ♣♦ss✉❛200❡s♣é❝✐❡s ❞❡ ❛♥✐♠❛✐s❄
✭❡✮ ❉❛❞❛s ❞✉❛s ✐❧❤❛s ❝♦♠ ár❡❛s ❞✐st✐♥t❛s✱ é ♣♦ssí✈❡❧ q✉❡ ❡❧❛s ♣♦ss✉❛♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ és♣❡❝✐❡s❄ ✭❢✮ ❙❛❜❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❡s♣é❝✐❡s ❡♠ ✉♠❛ ✐❧❤❛✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r s✉❛ ár❡❛❄
✻✳ ❯♠ ❝❛♥❤ã♦ é ❝♦❧♦❝❛❞♦ ♥❛ ♦r✐❣❡♠ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠ ♣r♦❥ét✐❧ ❛t✐r❛❞♦ ♣❡❧♦ ❝❛♥❤ã♦ t❡♠ ❝♦♦r❞❡♥❛❞❛s x= 50t ♠❡tr♦s ❡ y = 50t−t2 ♠❡tr♦s ❞❡♣♦✐s ❞❡ t s❡❣✉♥❞♦s ❞♦ ❧❛♥ç❛♠❡♥t♦✳ ▼♦str❡ q✉❡ ❛ tr❛❥❡tór✐❛ ❞♦ ♣r♦❥ét✐❧ é ✉♠❛ ♣❛rá❜♦❧❛✳ ❆ q✉❡ ❞✐stâ♥❝✐❛ ❞♦ ❝❛♥❤ã♦ ♦ ♣r♦❥ét✐❧ ✈❛✐ ❛t✐♥❣✐r ♦ ❝❤ã♦❄ ◗✉❛❧ ❛ ❛❧t✉r❛ ♠á①✐♠❛ q✉❡ ♦ ♣r♦❥ét✐❧ ✈❛✐ ❛t✐♥❣✐r❄
✼✳ ❯♠ ❤♦♠❡♠ ❞❡1,80♠❡tr♦s ❞❡ ❛❧t✉r❛ ❡stá ♣❛r❛❞♦✱ ❛♦ ♥í✈❡❧ ❞❛ r✉❛✱ ♣❡rt♦ ❞❡ ✉♠ ♣♦st❡ ❞❡ ✐❧✉♠✐♥❛çã♦ ❞❡4,50♠❡tr♦s
q✉❡ ❡stá ❛❝❡s♦✳ ❊①♣r✐♠❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ s✉❛ s♦♠❜r❛ ❝♦♠♦ ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛ q✉❡ ❡❧❡ ❡stá ❞♦ ♣♦st❡✳
✽✳ ❯♠ t❛♥q✉❡✱ ❝♦♠ á❣✉❛✱ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♥❡ ❝✐r❝✉❧❛r r❡t♦✱ ❝♦♠ ✈ért✐❝❡ ❛♣♦♥t❛♥❞♦ ♣❛r❛ ❜❛✐①♦✳ ❖ r❛✐♦ ❞❛ ❜❛s❡ ❞♦ ❝♦♥❡ é ✐❣✉❛❧ ❛9 ♠❡tr♦s ❡ s✉❛ ❛❧t✉r❛ é ❞❡27♠❡tr♦s✳ ❊①♣r✐♠❛ ♦ ✈♦❧✉♠❡ ❞❡ á❣✉❛ ♥♦ t❛♥q✉❡ ❝♦♠♦ ❢✉♥çã♦ ❞❡ s✉❛
♣r♦❢✉♥❞✐❞❛❞❡✳
✾✳ ❯♠ ♦❜❥❡t♦ é ❧❛♥ç❛❞♦✱ ✈❡rt✐❝❛❧♠❡♥t❡✱ ❡ s❛❜❡✲s❡ q✉❡ ♥♦ ✐♥st❛♥t❡ t s❡❣✉♥❞♦s s✉❛ ❛❧t✉r❛ é ❞❛❞❛ ♣♦r h(t) = 4t−t2 q✉✐❧ô♠❡tr♦s✱ 0≤t≤4.
✭❛✮ ❊s❜♦❝❡ ♦ ❣rá✜❝♦ ❞❡h=h(t)✳
