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Ðàö³îíàëüí³ ³ ³ððàö³îíàëüí³ ôóíêö³¿ âõîäÿòü äî á³ëüø<br />
çàãàëüíîãî êëàñó — àëãåáðà¿÷íèõ ôóíêö³é. Âñ³ ³íø³ åëåìåíòàðí³<br />
ôóíêö³¿ íàçèâàþòüñÿ òðàíñöåíäåíòíèìè.<br />
6.1.4. Äåÿê³ âàæëèâ³ êëàñè ôóíêö³é<br />
Ôóíêö³ÿ f(x) (x∈D(f)), ùî ìຠâëàñòèâ³ñòü f(x) =f(−x), íàçèâàºòüñÿ<br />
ïàðíîþ, íàïðèêëàä õ 2 , cos x, à ùî ìຠâëàñòèâ³ñòü<br />
f(x) =−f(−x) — íåïàðíîþ, íàïðèêëàä õ 3 , sin x. Áàãàòî ôóíêö³é<br />
º í³ ïàðíèìè, í³ íåïàðíèìè, íàïðèêëàä à õ , x .<br />
ßêùî ³ñíóº ä³éñíå ÷èñëî Ò > 0, ùî ïðè áóäü-ÿêèõ çíà÷åííÿõ<br />
àðãóìåíòó õ∈D(f) ìຠì³ñöå ð³âí³ñòü<br />
f(x) =f(x + kT), (6.1.1)<br />
äå k — ö³ëå ÷èñëî, òî ó = f(x) íàçèâàºòüñÿ ïåð³îäè÷íîþ<br />
ôóíêö³ºþ, à Ò — ïåð³îäîì.<br />
Ðîçãëÿíåìî ôóíêö³þ y = f(x), îáëàñòþ âèçíà÷åííÿ ³ çíà-<br />
÷åíü ÿêî¿ º â³äïîâ³äíî D(f) i E(f). Òîä³ áóäü-ÿêîìó çíà÷åííþ<br />
x0<br />
Î Df () â³äïîâ³äàòèìå îäíå çíà÷åííÿ y0<br />
Î Ef (), ÿêå äîð³âíþº<br />
f(x 0 ).<br />
Òàêèì ÷èíîì, ð³âíÿííÿ<br />
y = f(x)<br />
ïðè y0<br />
Î Ef () ìຠïðèíàéìí³ îäèí êîð³íü. ²íøèìè ñëîâàìè,<br />
ð³âíÿííÿ (6.1.1) êîæíîìó yÎ Ef () ñòàâèòü ó â³äïîâ³äí³ñòü<br />
îäíå àáî ê³ëüêà çíà÷åíü x0<br />
Î Df (). ßêùî ïðè öüîìó êîæíîìó<br />
yÎ Ef () â³äïîâ³äຠò³ëüêè îäíå çíà÷åííÿ x0<br />
Î Df (), òî êàæóòü,<br />
ùî íà ìíîæèí³ E(f) çàäàíî ôóíêö³þ x=j ( y)<br />
(áóêâà ϕ<br />
îçíà÷àº, ùî õàðàêòåðèñòèêà íîâî¿ ôóíêö³¿ ³íøà). Íîâó ôóíêö³þ<br />
x = ϕ(y) íàçèâàþòü îáåðíåíîþ ôóíêö³ºþ äî ôóíêö³¿<br />
y = f(x), à y = f(x) ïðÿìîþ ôóíêö³ºþ. Ïðè öüîìó ôóíêö³¿<br />
x = ϕ(y) ³ y = f(x) íàçèâàþòü âçàºìíî îáåðíåíèìè. Íàâåäåí³<br />
âèùå ì³ðêóâàííÿ ïîêàçóþòü, ùî îáëàñòü âèçíà÷åííÿ òà îáëàñòü<br />
çíà÷åíü âçàºìíî îáåðíåíèõ ôóíêö³é ì³íÿþòüñÿ ì³æ<br />
ñîáîþ. Íàïðèêëàä, f = à õ , D(f) =R, E(f) =R + ; ϕ = log a y,<br />
D(ϕ )=R + , E(ϕ )=R.<br />
Ãðàô³êè âçàºìíî îáåðíåíèõ ôóíêö³é çá³ãàþòüñÿ. ßêùî<br />
äëÿ çðó÷íîñò³ ìè ðîçãëÿíåìî ôóíêö³þ y = ϕ(x), òî ìîæíà<br />
ïîêàçàòè, ùî ¿¿ ãðàô³ê ñèìåòðè÷íèé äî ãðàô³êà ïðÿìî¿<br />
ôóíêö³¿ â³äíîñíî á³ñåêòðèñè ïåðøîãî ³ òðåòüîãî êîîðäèíàòíèõ<br />
êóò³â. Öå òâåðäæåííÿ íàâîäèòüñÿ â êóðñ³ åëåìåíòàðíî¿<br />
ìàòåìàòèêè.<br />
ßêùî äëÿ äâîõ áóäü-ÿêèõ ð³çíèõ çíà÷åíü àðãóìåíòó õ 1 ³<br />
õ 2 , ÿê³ íàëåæàòü îáëàñò³ âèçíà÷åííÿ D(f), ³ç íåð³âíîñò³<br />
x 1 < õ 2 âèïëèâàº:<br />
1) f(x 1 )f(x 2 ) — ñïàäíîþ;<br />
4) f(x 1 ) ≥ f(x 2 ) — íåçðîñòàþ÷îþ.<br />
Òàê³ ôóíêö³¿ íàçèâàþòüñÿ ìîíîòîííèìè. Ïðè öüîìó<br />
ôóíêö³¿, ÿê³ çàäîâîëüíÿþòü âèìîãàì 1) àáî 3), íàçèâàþòüñÿ<br />
ñòðîãî ìîíîòîííèìè.<br />
6.1.5. Ïîáóäîâà ãðàô³ê³â ôóíêö³é ìåòîäîì ïåðåòâîðåíü<br />
Âèâ÷åííÿ âèùî¿ ìàòåìàòèêè ïðèïóñêຠäîáð³ çíàííÿ âëàñòèâîñòåé<br />
îñíîâíèõ åëåìåíòàðíèõ ôóíêö³é. ßê áóëî âæå<br />
ñêàçàíî, äî íèõ â³äíîñÿòüñÿ ñòàëà, ñòåïåíåâà, ïîêàçíèêîâà,<br />
ëîãàðèôì³÷íà, òðèãîíîìåòðè÷í³ ³ îáåðíåí³ òðèãîíîìåòðè÷í³<br />
ôóíêö³¿. Ïîâòîðèòè ¿õ âëàñòèâîñò³ ç â³äïîâ³äíîþ ïîáóäîâîþ<br />
ãðàô³ê³â ìîæíà ïî øê³ëüíèì ï³äðó÷íèêàì.<br />
Ïðè ïîäàëüøîìó âèâ÷åíí³ òåìè, ÿêà ïîâ’ÿçàíà ç ïîáóäîâîþ<br />
ãðàô³ê³â ôóíêö³é, âåëüìè êîðèñíèìè áóäóòü áåçïîñåðåäíüî<br />
ïåðåâ³ðÿºì³ òàê³ òâåðäæåííÿ:<br />
1. Ãðàô³ê ôóíêö³¿ y = −f(x) ñèìåòðè÷íèé ãðàô³êó ôóíêö³¿<br />
y = f(x) â³äíîñíî â³ñ³ Îõ.<br />
2. Ãðàô³ê ôóíêö³¿ y = f( x)<br />
ñï³âïàäຠç ãðàô³êîì ôóíêö³¿<br />
y = f(x) â òèõ òî÷êàõ, äå f(x) ≥ 0, ³ ñèìåòðè÷íèé éîìó â³äíîñíî<br />
â³ñ³ Îx â òî÷êàõ, äå f(x) 0, àáî íà b îäèíèöü ìàñøòàáó<br />
âíèç, ÿêùî b
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