Функция одной переменной (Конспект), страница 4

Ôóíêöèÿ íåîãðàíè÷åííàÿ, íåïåðèîäè÷åñêàÿ. Ôóíêöèÿ íè ÷¼òíàÿ, íè íå÷¼òíàÿ. Ôóíêöèÿ ïðè a > 1 âîçðàñòàåò, ïðè a < 1óáûâàåò. Òî÷êà ïåðåñå÷åíèÿ ñ îñüþ Ox : (1; 0). Ãðàôèê ôóíêöèè ïðèxâåäåí íà ðèñ. 40. Ôóíêöèÿ îáðàòèìàÿ, îáðàòíàÿ ôóíêöèÿ y = a .Ïîñêîëüêó ôóíêöèÿ4.8. Òðèãîíîìåòðè÷åñêèå ôóíêöèèy = sin xD(f ) = (−∞; +∞), E(f ) = [−1; 1].Ôóíêöèÿ îãðàíè÷åíà: −1 6 sin x 6 1. Ôóíêöèÿ íå÷¼òíàÿ, ò.ê. D(f )ñèììåòðè÷íà îòíîñèòåëüíî O è f (−x) = sin(−x) = − sin x = −f (x).Ôóíêöèÿ ïåðèîäè÷åñêàÿ ñ ïåðèîäîì T = 2π . Ôóíêöèÿ íå ìîíîòîííàÿ:πππ3πâîçðàñòàåò íà (− +2πn; +2πn), n ∈ Z; óáûâàåò íà ( +2πn;+2πn),2222ãäå n ∈ Z.

Òî÷êè ïåðåñå÷åíèÿ ñ îñüþ Ox : (πn; 0), n ∈ Z, ñ îñüþOy : (0; 0). Ãðàôèêîì ôóíêöèè ÿâëÿåòñÿ ñèíóñîèäà (ðèñ. 41). Ôóíêπ πöèÿ íåîáðàòèìàÿ, íî åñëè ðàññìîòðåòü å¼ íà [− ; ], òî ñóùåñòâóåò2 2îáðàòíàÿ ôóíêöèÿ y = arcsin x.y = cos x.D(f ) = (−∞; +∞), E(f ) = [−1; 1].Ôóíêöèÿ îãðàíè÷åíà −1 6 cos x 6 1.

Ôóíêöèÿ ÷¼òíàÿ, ò.ê. D(f ) ñèììåòðè÷íà îòíîñèòåëüíî O è f (−x) = cos(−x) = cos x = f (x). Ôóíêöèÿïåðèîäè÷åñêàÿ ñ ïåðèîäîì T = 2π .Ôóíêöèÿ íå ìîíîòîííàÿ: óáûâàåò íà (2πn; π + 2πn) è âîçðàñòàåòíà (π + 2πn; 2π + 2πn), n ∈ Z.πÒî÷êè ïåðåñå÷åíèÿ ñ îñüþ Ox: ( +πn; 0); n ∈ Z òî÷êà ïåðåñå÷åíèÿ2ñ îñüþ Oy: (0;1).Ãðàôèê ôóíêöèèy = cos xèçîáðàæåí íà ðèñ. 42.Ôóíêöèÿ íåîáðàòèìàÿ, íî åñëè ðàññìîòðåòü å¼ íàñòâóåò îáðàòíàÿ ôóíêöèÿ[0; π],òî ñóùå-y = arccos x.y = tg x.D(f ) = {x| x ∈ R, x ̸=π+ πn, ãäå n ∈ Z}; E(f ) = (−∞; +∞).2Ôóíêöèÿ íåîãðàíè÷åííàÿ.Ôóíêöèÿ íå÷¼òíàÿ, ò.ê.

D(f ) ñèììåòðè÷íà îòíîñèòåëüíî Î è f (−x) == tg(−x) = − tg x = −f (x).Ëåêöèÿ 4. Îñíîâíûå ýëåìåíòàðíûå ôóíêöèè65yy1π1π2xπ2Oπ−π-1π2π2xπO-1y=cosxÐèñ. 41.ÃðàôèêÐèñ. 42.Ãðàôèêôóíêöèèy = sin xôóíêöèèy = cos xT = π.ππÔóíêöèÿ íå ìîíîòîííàÿ; âîçðàñòàåò íà (− + πn;+ πn), n ∈ Z.22Òî÷êè ïåðåñå÷åíèÿ ñ îñüþ Ox: (πn; 0); n ∈ Z, òî÷êà ïåðåñå÷åíèÿ ñÔóíêöèÿ ïåðèîäè÷åñêàÿ ñ ïåðèîäîìîñüþ Oy: (0;0).Ãðàôèê ôóíêöèèy = tg xèçîáðàæåí íà ðèñ. 43Ôóíêöèÿ íåîáðàòèìàÿ, íî åñëè ðàññìîòðåòü å¼ íàùåñòâóåò îáðàòíàÿ ôóíêöèÿy = arctg x.(− π2 ; π2 ),òî ñó-y = ctg x.D(f ) = {x| x ∈ R, x ̸= πn, ãäå n ∈ Z}; E(f ) = (−∞; +∞).Ôóíêöèÿ íåîãðàíè÷åííàÿ.D(f ) ñèììåòðè÷íà îòíîñèòåëüíî Î è f (−x) == ctg(−x) = − ctg x = −f (x).Ôóíêöèÿ ïåðèîäè÷åñêàÿ ñ ïåðèîäîì T = π.Ôóíêöèÿ íå ìîíîòîííàÿ; óáûâàåò íà (πn; π + πn), n ∈ Z.πÒî÷êè ïåðåñå÷åíèÿ ñ îñüþ Ox: ( + πn; 0) n ∈ Z.2Ãðàôèê ôóíêöèè y = ctg x èçîáðàæåí íà ðèñ.

44.Ôóíêöèÿ íåîáðàòèìàÿ, íî åñëè ðàññìîòðåòü å¼ íà (0; π), òî ñóùåñòâóåò îáðàòíàÿ ôóíêöèÿ y = arcctg x.Ôóíêöèÿ íå÷¼òíàÿ, ò.ê.4.9. Îáðàòíûå òðèãîíîìåòðè÷åñêèå ôóíêöèèÑâîéñòâà ýòèõ ôóíêöèé ïîëó÷àþòñÿ èç ñâîéñòâ òðèãîíîìåòðè÷åñêèõ ôóíêöèé íà îñíîâàíèè ï. 3.8y = arcsin x.[ π π].D(f ) = [−1; 1]; E(f ) = − ;2 266Ëåêöèÿ 4. Îñíîâíûå ýëåìåíòàðíûå ôóíêöèèy−3π/2−π−π/2 0y−ππ/2πÐèñ. 43.Ãðàôèêôóíêöèèy = tg xx−π/20π/2πÐèñ.

44.Ãðàôèêôóíêöèèy = ctg xx− π2 6 arcsin x 6 π2 .arcsin(−x) = − arcsin x.Ôóíêöèÿ îãðàíè÷åíà:Ôóíêöèÿ íå÷¼òíàÿ:Ôóíêöèÿ íåïåðèîäè÷åñêàÿ.Ôóíêöèÿ âîçðàñòàåò íà îáëàñòè îïðåäåëåíèÿ.Òî÷êà ïåðåñå÷åíèÿ ñ îñÿìè: (0;0).Ãðàôèê ôóíêöèè èçîáðàæåí íà ðèñ. 45.y = arccos x.D(f ) = [−1; 1]; E(f ) = [0; π].Ôóíêöèÿ îãðàíè÷åíà:arccos(−x) = π − arccos x.0 6 arccos x 6 π.Ìîæíî ïîêàçàòü, ÷òîÔóíêöèÿ óáûâàåò íà îáëàñòè îïðåäåëåíèÿ.Òî÷êà ïåðåñå÷åíèÿ ñ îñüþ Ox: (1;0); ñ îñüþ Oy:(0; π2 ).Ãðàôèê ôóíêöèè èçîáðàæåí íà ðèñ. 46.y = arctg x.ÔóíêöèÿÔóíêöèÿ( π π)D(f ) = (−∞; +∞) ; E(f ) = − ;.2 2ππîãðàíè÷åíà: − < arctg x < .22íå÷¼òíàÿ: arctg(−x) = − arctg xÔóíêöèÿ íåïåðèîäè÷åñêàÿ.Ôóíêöèÿ âîçðàñòàåò íà îáëàñòè îïðåäåëåíèÿ.Òî÷êà ïåðåñå÷åíèÿ ñ îñÿìè: (0;0).Ãðàôèê ôóíêöèè èçîáðàæåí íà ðèñ. 47.Ëåêöèÿ 4.

Îñíîâíûå ýëåìåíòàðíûå ôóíêöèè67yyππ/20-11π/2x−π/2-1Ðèñ. 45.Ãðàôèêôóíêöèèy = arcsin x0x1Ðèñ. 46.Ãðàôèêôóíêöèèy = arccos xy = arcctg x.( π)D(f ) = (−∞; +∞) ; E(f ) = 0;.2Ôóíêöèÿ îãðàíè÷åíà: 0 < arcctg x < π .Ìîæíî ïîêàçàòü, ÷òî arcctg(−x) = π − arcctg x.Ôóíêöèÿ óáûâàåò íà îáëàñòè îïðåäåëåíèÿ.πÒî÷êà ïåðåñå÷åíèÿ ñ îñüþ Oy: (0; ).2Ãðàôèê ôóíêöèè èçîáðàæåí íà ðèñ.

48.yyππ/2π/2OxO−π/2xÐèñ. 47.ÃðàôèêÐèñ. 48.ôóíêöèèy = arctg xôóíêöèèÃðàôèêy = arcctg x68Ëåêöèÿ 4. Îñíîâíûå ýëåìåíòàðíûå ôóíêöèè4.10. Ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ ãðàôèêîâÃðàôèêè ìíîãèõ ôóíêöèé ìîæíî ïîëó÷èòü èç ðàíåå ðàññìîòðåííûõ ñ ïîìîùüþ ýëåìåíòàðíûõ ãåîìåòðè÷åñêèõ ïðåîáðàçîâàíèé: ïàðàëëåëüíîãî ïåðåíîñà, ñæàòèÿ, ðàñòÿæåíèÿ, ñèììåòðè÷íîãî îòîáðàæåíèÿ. Ðàññìîòðèì íåêîòîðûå èç ýòèõ ïðåîáðàçîâàíèé. Äëÿ êàæäîãîèç ýëåìåíòàðíûõ ïðåîáðàçîâàíèé ïðåäëàãàåòñÿ äâà ñïîñîáà ïîñòðîåíèÿ ãðàôèêà: ñ ïîìîùüþ ïðåîáðàçîâàíèÿ ãðàôèêà è ñ ïîìîùüþ ïðåîáðàçîâàíèÿ ñèñòåìû êîîðäèíàò. Îáó÷àþùèéñÿ äîëæåí âûáðàòü òîò,êîòîðûé êàæåòñÿ åìó ïðîùå, è îâëàäåòü èì.

 êàæäîì ñëó÷àå ñ÷èòàåòñÿ èçâåñòíûì ãðàôèê ôóíêöèèy = f (x).4.11. Ïàðàëëåëüíûé ïåðåíîñ ãðàôèêîâÃðàôèê ôóíêöèèy = f (x)y = f (x) + bñëåäóþùèì îáðàçîì: êî âñåì îðäèíàòàì ãðàôèêàïðèáàâëÿåòñÿ âåëè÷èíàÅñëèïîëó÷àåòñÿ èç ãðàôèêà ôóíêöèèb,y = f (x)Oy .÷òî îçíà÷àåò ñäâèã ãðàôèêà âäîëü îñèb > 0, òî ãðàôèê ôóíêöèè y = f (x) ïåðåíîñèòñÿ ââåðõ ïàðàëëåëüOy íà b, åñëè b < 0, òî ãðàôèê ôóíêöèè y = f (x) ïåðåíîñèòñÿïàðàëëåëüíî îñè Oy íà |b| (ðèñ. 49). Çàìåòèì, ÷òî âìåñòî ïå-íî îñèâíèçðåíîñà ãðàôèêà, ìîæíî ïåðåíåñòè â ïðîòèâîïîëîæíîì íàïðàâëåíèèîñüOx(åñëèb > 0 âíèç, åñëè b < 0Oy âåëè÷èíó b.

ââåðõ), ïðèáàâèâ êî âñåìçíà÷åíèÿì ïî îñèyy=f(x)+by=f(x)b>0b0Ðèñ. 49.xÏîñòðîåíèå ãðàôèêà ôóíêöèèy = f (x) + bÃðàôèê ôóíêöèè y = x2 − 1 (ðèñ. 50) ñìåù¼í íà 1âíèç ïàðàëëåëüíî îñè Oy îòíîñèòåëüíî ãðàôèêà ôóíêöèè y = x2 .Ïðèìåð 4.1.y = f (x + a) ïîëó÷àåòñÿ ñ ïîìîùüþ ïåðåíîñày = f (x). Äåéñòâèòåëüíî, ïåðåéäÿ ê íîâûì êîîðäèX = x + a, Y = y ïàðàëëåëüíûì ïåðåíîñîì âäîëü îñè Ox íàÃðàôèê ôóíêöèèãðàôèêà ôóíêöèèíàòàì−a, çàìåòèì, ÷òî îòíîñèòåëüíî íîâûõ êîîðäèíàò ïîëó÷èòñÿ èñõîäíûéËåêöèÿ 4.

Îñíîâíûå ýëåìåíòàðíûå ôóíêöèè692yy=x2y=x -10x-11-1Ðèñ. 50.ãðàôèê ôóíêöèèÏîñòðîåíèå ãðàôèêà ôóíêöèèY = f (X).Åñëèa > 0,y = x2 − 1òî ñòàðûå êîîðäèíàòû ïîëó-÷àþòñÿ èç íîâûõ ñäâèãîì íàïðàâî âäîëü îñèOxíàa,ò.ê.x = X − a.Åñëè æå ñäâèãàòü ãðàôèê, à íå ñèñòåìó êîîðäèíàò, òî åãî íóæíî äâè-a > 0, òîãðàôèê ôóíêöèè y = f (x) ïåðåíîñèòñÿ íàëåâî ïàðàëëåëüíî îñè Ox íàa, åñëè a < 0, òî ãðàôèê ôóíêöèè y = f (x) ïåðåíîñèòñÿ íàïðàâî âäîëüîñè Ox íà |a| (ðèñ. 51). Âìåñòî ãðàôèêà ìîæíî ïåðåíåñòè â ïðîòèâîïîëîæíîì íàïðàâëåíèè îñü Oy (åñëè a > 0 âïðàâî, åñëè a < 0 âëåâî), îòíÿâ îò âñåõ çíà÷åíèé ïî îñè Ox âåëè÷èíó a.ãàòü â ïðîòèâîïîëîæíîì íàïðàâëåíèè íàëåâî. Èòàê, åñëèy=f(x+a)yy=f(x)a>0a0Ðèñ.

51.xÏîñòðîåíèå ãðàôèêà ôóíêöèèy = f (x + a)Ãðàôèê ôóíêöèè y = (x − 2)2 ñìåù¼í íà 2 åä. âïðàâîïàðàëëåëüíî îñè Îõ îòíîñèòåëüíî ãðàôèêà ôóíêöèè y = x2 . (ðèñ. 52).Ïðèìåð 4.2.4.12. Ñæàòèå è ðàñòÿæåíèå ãðàôèêîâk ∈ R, ïîëó÷àåòñÿ ñ ïîìîùüþy = f (x) â k ðàç â íàïðàâëåíèè îòîñè Ox. ¾Ðàñòÿæåíèå¿ çäåñü ïîíèìàåòñÿ êàê óìíîæåíèå íà k îðäèíàòâñåõ òî÷åê ãðàôèêà y = f (x).

Ïðè k > 1 ýòî áóäåò äåéñòâèòåëüíîÃðàôèê ôóíêöèèy = kf (x),¾ðàñòÿæåíèÿ¿ ãðàôèêà ôóíêöèèãäå70Ëåêöèÿ 4. Îñíîâíûå ýëåìåíòàðíûå ôóíêöèèy2y=xy=(x-2)2x0Ðèñ. 52.2Ïîñòðîåíèå ãðàôèêà ôóíêöèèy = (x − 2)2ðàç îò îñè Ox âäîëü îñè Oy . Ïðè 0 < k < 1 ýòî1ðàç ê îñè Ox âäîëü îñè Oy .

Ïðè k 6 −1 ýòî áóäåòkðàñòÿæåíèå â |k| ðàç ñ ïîñëåäóþùèì ñèììåòðè÷íûì îòîáðàæåíèåìðàñòÿæåíèå âkáóäåò ñæàòèå âîòíîñèòåëüíî îñè Ox (ïåðåâåðíóòü ñâåðõó âíèç); ïðè −1 6 k < 0 ýòî1áóäåò ñæàòèå âðàç è ñèììåòðèÿ îòíîñèòåëüíî îñè Ox (ðèñ. 53).|k|y = −f (x) ïîëó÷àåòñÿ ñèììåòðè÷íûìOx ãðàôèêà ôóíêöèè y = f (x).Âìåñòî ïðåîáðàçîâàíèÿ ãðàôèêà ïðè k > 0 ìîæíî èñïðàâèòü çíà÷åíèÿ ïî îñè Oy , óìíîæèâ èõ íà k . Ïðè k < 0 â ýòîì ñëó÷àå ïðèøëîñü ÷àñòíîñòè, ãðàôèê ôóíêöèèîòîáðàæåíèåì îòíîñèòåëüíî îñèáû ìåíÿòü íàïðàâëåíèå îñè, ÷òî íåóäîáíî; ëó÷øå ïåðåâåðíóòü ãðàôèêñâåðõó âíèç.y- π2- 2π-1-23210y=-3sin xπ2πy=sin x2π x-3Ðèñ.

53.Ïîñòðîåíèå ãðàôèêà ôóíêöèèy = −3 sin xy = f (kx), ãäå k ∈ R, ïîëó÷àåòñÿ ñ ïîìîùüþy = f (x) â k ðàç â íàïðàâëåíèè ê îñè Oy . ¾Ñæàòèå¿ çäåñü ïîíèìàåòñÿ êàê äåëåíèå íà k àáñöèññ âñåõ òî÷åê ãðàôèêày = f (x). Äåéñòâèòåëüíî, åñëè, íàïðèìåð, f (1) = 0, òî, ñäåëàâ çàìåíóÃðàôèê ôóíêöèè¾ñæàòèÿ¿ ãðàôèêàËåêöèÿ 4.

Îñíîâíûå ýëåìåíòàðíûå ôóíêöèè71X = kx, Y = y, ïîëó÷èì, ÷òî ôóíêöèÿ y = f (kx) îáðàùàåòñÿ â íóëü1ïðè kx = 1, ò.å. ïðè x = .kÏðè k > 1 ãðàôèê ôóíêöèè y = f (x) ñæèìàåòñÿ â k ðàç ê îñè Oyâäîëü îñè Ox; ïðè 0 < k < 1 ãðàôèê ôóíêöèè y = f (x) ðàñòÿãèâàåòñÿ1âðàç îò îñè Oy âäîëü îñè Ox; ïðè k 6 −1 èñõîäíûé ãðàôèê ñæèìàkåòñÿ â |k| ðàç è ñèììåòðè÷íî îòðàæàåòñÿ îòíîñèòåëüíî îñè Oy (ñëåâà1íàïðàâî); ïðè −1 6 k < 0 èñõîäíûé ãðàôèê ðàñòÿãèâàåòñÿ âðàç ñ|k|ïîñëåäóþùåé ñèììåòðèåé îòíîñèòåëüíî îñè Oy . ÷àñòíîñòè, ãðàôèê ôóíêöèè y = f (−x) ïîëó÷àåòñÿ èç ãðàôèêàôóíêöèè y = f (x) ñèììåòðèåé îòíîñèòåëüíî îñè Oy .Âìåñòî ïðåîáðàçîâàíèÿ ãðàôèêà ïðè k > 0 ìîæíî èñïðàâèòü çíà÷åíèÿ ïî îñè Ox, ïîäåëèâ èõ íà k .

Ïðè k < 0 â ýòîì ñëó÷àå ñëåäóåòïðåäâàðèòåëüíî ïåðåâåðíóòü ãðàôèê ñëåâà íàïðàâî.Ãðàôèê ôóíêöèè y = cos 2x ïîëó÷àåòñÿ èç ãðàôèêày = cos x ñæàòèåì â 2 ðàçà ê îñè Îy; ãðàôèê ôóíêöèè y = ln(−x)ïîëó÷àåòñÿ èç ãðàôèêà y = ln x ñèììåòðèåé îòíîñèòåëüíî îñè Oy(ðèñ. 54).Ïðèìåð 4.3.yy=ln(-x)y=ln x0Ðèñ. 54.x1-1Ïîñòðîåíèå ãðàôèêà ôóíêöèèy = ln(−x)Ïîëüçóÿñü èçëîæåííûìè ìåòîäàìè, ïðèâåäåì ïîñëåäîâàòåëüíîñòüïðåîáðàçîâàíèé ïðè ïîñòðîåíèè ãðàôèêà ôóíêöèèäàí ãðàôèê ôóíêöèè•••y = f (kx + b), åñëèy = f (x):y = f (x);y = f (x + b),íàðèñîâàòü ãðàôèê ôóíêöèèïîëó÷èòü ãðàôèê ôóíêöèèñäâèíóâ èñõîäíûé,êàê îïèñàíî â ï. 4.11;ïîëó÷èòü ãðàôèê ôóíêöèèâkðàç ê îñèOy ,y = f (kx+b), ¾ñæàâ¿ ïðåäûäóùèéêàê îïèñàíî âûøå.72Ïðàêòè÷åñêîå çàíÿòèå 3.

Îñíîâíûå ñâîéñòâà ôóíêöèéÏðèìåð 4.4. Íàïèñàòü ïîñëåäîâàòåëüíîñòü ïðåîáðàçîâàíèé è ïî√ñòðîèòü ãðàôèê ôóíêöèè y = 4 − 5x.Ð å ø å í è å:√y = √x;ïîëó÷èì ãðàôèê ôóíêöèè y =x + 4, ñäâèíóâ èñõîäíûé íà4 åäèíèöû âëåâî âäîëü îñè Ox;√• ïîëó÷èì ãðàôèê ôóíêöèè y = −5x + 4, ñæàâ ïðåäûäóùèé â5 ðàç ê îñè Oy è çàòåì îòîáðàçèâ ñèììåòðè÷íî îòíîñèòåëüíîîñè Oy .••íàðèñóåì ãðàôèê ôóíêöèèÏîñòðîåíèå ãðàôèêà ïîêàçàíî íà ðèñ.