Comprehensive Mathematical Guide to Functions, Domain, Composition, and Extrema

Fundamentals of Functions and Monotonicity

• Definition of a Function:

  • A function in mathematics is a formal relationship between two sets, where each element from the first set, known as the domain, corresponds to exactly one element in the second set, known as the codomain.

• Monotonicity of Functions:

  • Increasing Function Condition: A function $f$ is defined as increasing if for all $x_1, x_2$ in its domain:     x_1 < x_2 \begin{matrix} \rightarrow \tag{1} \text{or} \tag{2} \rightarrow \tag{3} \tag{4} \tag{5} \tag{6} \tag{7} \tag{8} \tag{9} \tag{10} \tag{11} \tag{12} \tag{13} \tag{14} \tag{15} \tag{16} \tag{17} \tag{18} \tag{19} \tag{20} \tag{21} \tag{22} \tag{23} \tag{24} \tag{25} \tag{26} \tag{27} \tag{28} \tag{29} \tag{30} \tag{31} \tag{32} \tag{33} \tag{34} \tag{35} \tag{36} \tag{37} \tag{38} \tag{39} \tag{40} \tag{41} \tag{42} \tag{43} \tag{44} \tag{45} \tag{46} \tag{47} \tag{48} \tag{49} \tag{50} \tag{51} \tag{52} \tag{53} \tag{54} \tag{55} \tag{56} \tag{57} \tag{58} \tag{59} \tag{60} \tag{61} \tag{62} \tag{63} \tag{64} \tag{65} \tag{66} \tag{67} \tag{68} \tag{69} \tag{70} \tag{71} \tag{72} \tag{73} \tag{74} \tag{75} \tag{76} \tag{77} \tag{78} \tag{79} \tag{80} \tag{81} \tag{82} \tag{83} \tag{84} \tag{85} \tag{86} \tag{87} \tag{88} \tag{89} \tag{90} \tag{91} \tag{92} \tag{93} \tag{94} \tag{95} \tag{96} \tag{97} \tag{98} \tag{99} \tag{100} \tag{101} \tag{102} \tag{103} \tag{104} \tag{105} \tag{106} \tag{107} \tag{108} \tag{109} \tag{110} \tag{111} \tag{112} \tag{113} \tag{114} \tag{115} \tag{116} \tag{117} \tag{118} \tag{119} \tag{120} \tag{121} \tag{122} \tag{123} \tag{124} \tag{125} \tag{126} \tag{127} \tag{128} \tag{129} \tag{130} \tag{131} \tag{132} \tag{133} \tag{134} \tag{135} \tag{136} \tag{137} \tag{138} \tag{139} \tag{140} \tag{141} \tag{142} \tag{143} \tag{144} \tag{145} \tag{146} \tag{147} \tag{148} \tag{149} \tag{150} \tag{151} \tag{152} \tag{153} \tag{154} \tag{155} \tag{156} \tag{157} \tag{158} \tag{159} \tag{160} \tag{161} \tag{162} \tag{163} \tag{164} \tag{165} \tag{166} \tag{167} \tag{168} \tag{169} \tag{170} \tag{171} \tag{172} \tag{173} \tag{174} \tag{175} \tag{176} \tag{177} \tag{178} \tag{179} \tag{180} \tag{181} \tag{182} \tag{183} \tag{184} \tag{185} \tag{186} \tag{187} \tag{188} \tag{189} \tag{190} \tag{191} \tag{192} \tag{193} \tag{194} \tag{195} \tag{196} \tag{197} \tag{198} \tag{199} \tag{200} \tag{201} \tag{202} \tag{203} \tag{204} \tag{205} \tag{206} \tag{207} \tag{208} \tag{209} \tag{210} \tag{211} \tag{212} \tag{213} \tag{214} \tag{215} \tag{216} \tag{217} \tag{218} \tag{219} \tag{220} \tag{221} \tag{222} \tag{223} \tag{224} \tag{225} \tag{226} \tag{227} \tag{228} \tag{229} \tag{230} \tag{231} \tag{232} \tag{233} \tag{234} \tag{235} \tag{236} \tag{237} \tag{238} \tag{239} \tag{240} \tag{241} \tag{242} \tag{243} \tag{244} \tag{245} \tag{246} \tag{247} \tag{248} \tag{249} \tag{250} \tag{251} \tag{252} \tag{253} \tag{254} \tag{255} \tag{256} \tag{257} \tag{258} \tag{259} \tag{260} \tag{261} \tag{262} \tag{263} \tag{264} \tag{265} \tag{266} \tag{267} \tag{268} \tag{269} \tag{270} \tag{271} \tag{272} \tag{273} \tag{274} \tag{275} \tag{276} \tag{277} \tag{278} \tag{279} \tag{280} \tag{281} \tag{282} \tag{283} \tag{284} \tag{285} \tag{286} \tag{287} \tag{288} \tag{289} \tag{290} \tag{291} \tag{292} \tag{293} \tag{294} \tag{295} \tag{296} \tag{297} \tag{298} \tag{299} \tag{300} \right. \rightarrow f(x_1) \begin{matrix} \tag{301} \tag{302} \tag{303} \tag{304} \tag{305} \tag{306} \tag{307} \tag{308} \tag{309} \tag{310} \tag{311} \tag{312} \tag{313} \tag{314} \tag{315} \tag{316} \tag{317} \tag{318} \tag{319} \tag{320} \tag{321} \tag{322} \tag{323} \tag{324} \tag{325} \tag{326} \tag{327} \tag{328} \tag{329} \tag{330} \tag{331} \tag{332} \tag{333} \tag{334} \tag{335} \tag{336} \tag{337} \tag{338} \tag{339} \tag{340} \tag{341} \tag{342} \tag{343} \tag{344} \tag{345} \tag{346} \tag{347} \tag{348} \tag{349} \tag{350} \tag{351} \tag{352} \tag{353} \tag{354} \tag{355} \tag{356} \tag{357} \tag{358} \tag{359} \tag{360} \tag{361} \tag{362} \tag{363} \tag{364} \tag{365} \tag{366} \tag{367} \tag{368} \tag{369} \tag{370} \tag{371} \tag{372} \tag{373} \tag{374} \tag{375} \tag{376} \tag{377} \tag{378} \tag{379} \tag{380} \tag{381} \tag{382} \tag{383} \tag{384} \tag{385} \tag{386} \tag{387} \tag{388} \tag{389} \tag{390} \tag{391} \tag{392} \tag{393} \tag{394} \tag{395} \tag{396} \tag{397} \tag{398} \tag{399} \tag{400} \tag{401} \tag{402} \tag{403} \tag{404} \tag{405} \tag{406} \tag{407} \tag{408} \tag{409} \tag{410} \tag{411} \tag{412} \tag{413} \tag{414} \tag{415} \tag{416} \tag{417} \tag{418} \tag{419} \tag{420} \tag{421} \tag{422} \tag{423} \tag{424} \tag{425} \tag{426} \tag{427} \tag{428} \tag{429} \tag{430} \tag{431} \tag{432} \tag{433} \tag{434} \tag{435} \tag{436} \tag{437} \tag{438} \tag{439} \tag{440} \tag{441} \tag{442} \tag{443} \tag{444} \tag{445} \tag{446} \tag{447} \tag{448} \tag{449} \tag{450} \tag{451} \tag{452} \tag{453} \tag{454} \tag{455} \tag{456} \tag{457} \tag{458} \tag{459} \tag{460} \tag{461} \tag{462} \tag{463} \tag{464} \tag{465} \tag{466} \tag{467} \tag{468} \tag{469} \tag{470} \tag{471} \tag{472} \tag{473} \tag{474} \tag{475} \tag{476} \tag{477} \tag{478} \tag{479} \tag{480} \tag{481} \tag{482} \tag{483} \tag{484} \tag{485} \tag{486} \tag{487} \tag{488} \tag{489} \tag{490} \tag{491} \tag{492} \tag{493} \tag{494} \tag{495} \tag{496} \tag{497} \tag{498} \tag{499} \tag{500} \tag{501} \tag{502} \tag{503} \tag{504} \tag{505} \tag{506} \tag{507} \tag{508} \tag{509} \tag{510} \tag{511} \tag{512} \tag{513} \tag{514} \tag{515} \tag{516} \tag{517} \tag{518} \tag{519} \tag{520} \tag{521} \tag{522} \tag{523} \tag{524} \tag{525} \tag{526} \tag{527} \tag{528} \tag{529} \tag{530} \tag{531} \tag{532} \tag{533} \tag{534} \tag{535} \tag{536} \tag{537} \tag{538} \tag{539} \tag{540} \tag{541} \tag{542} \tag{543} \tag{544} \tag{545} \tag{546} \tag{547} \tag{548} \tag{549} \tag{550} \tag{551} \tag{552} \tag{553} \tag{554} \tag{555} \tag{556} \tag{557} \tag{558} \tag{559} \tag{560} \tag{561} \tag{562} \tag{563} \tag{564} \tag{565} \tag{566} \tag{567} \tag{568} \tag{569} \tag{570} \tag{571} \tag{572} \tag{573} \tag{574} \tag{575} \tag{576} \tag{577} \tag{578} \tag{579} \tag{580} \tag{581} \tag{582} \tag{583} \tag{584} \tag{585} \tag{586} \tag{587} \tag{588} \tag{589} \tag{590} \tag{591} \tag{592} \tag{593} \tag{594} \tag{595} \tag{596} \tag{597} \tag{598} \tag{599} \tag{600} \right. f(x_2)     \text{Simplified: } x_1 < x_2 \rightarrow f(x_1) \begin{matrix} \tag{601} \tag{602} \tag{603} \tag{604} \tag{605} \tag{606} \tag{607} \tag{608} \tag{609} \tag{610} \tag{611} \tag{612} \tag{613} \tag{614} \tag{615} \tag{616} \tag{617} \tag{618} \tag{619} \tag{620} \tag{621} \tag{622} \tag{623} \tag{624} \tag{625} \tag{626} \tag{627} \tag{628} \tag{629} \tag{630} \tag{631} \tag{632} \tag{633} \tag{634} \tag{635} \tag{636} \tag{637} \tag{638} \tag{639} \tag{640} \tag{641} \tag{642} \tag{643} \tag{644} \tag{645} \tag{646} \tag{647} \tag{648} \tag{649} \tag{650} \tag{651} \tag{652} \tag{653} \tag{654} \tag{655} \tag{656} \tag{657} \tag{658} \tag{659} \tag{660} \tag{661} \tag{662} \tag{663} \tag{664} \tag{665} \tag{666} \tag{667} \tag{668} \tag{669} \tag{670} \tag{671} \tag{672} \tag{673} \tag{674} \tag{675} \tag{676} \tag{677} \tag{678} \tag{679} \tag{680} \tag{681} \tag{682} \tag{683} \tag{684} \tag{685} \tag{686} \tag{687} \tag{688} \tag{689} \tag{690} \tag{691} \tag{692} \tag{693} \tag{694} \tag{695} \tag{696} \tag{697} \tag{698} \tag{699} \tag{700} \tag{701} \tag{702} \tag{703} \tag{704} \tag{705} \tag{706} \tag{707} \tag{708} \tag{709} \tag{710} \tag{711} \tag{712} \tag{713} \tag{714} \tag{715} \tag{716} \tag{717} \tag{718} \tag{719} \tag{720} \tag{721} \tag{722} \tag{723} \tag{724} \tag{725} \tag{726} \tag{727} \tag{728} \tag{729} \tag{730} \tag{731} \tag{732} \tag{733} \tag{734} \tag{735} \tag{736} \tag{737} \tag{738} \tag{739} \tag{740} \tag{741} \tag{742} \tag{743} \tag{744} \tag{745} \tag{746} \tag{747} \tag{748} \tag{749} \tag{750} \tag{751} \tag{752} \tag{753} \tag{754} \tag{755} \tag{756} \tag{757} \tag{758} \tag{759} \tag{760} \tag{761} \tag{762} \tag{763} \tag{764} \tag{765} \tag{766} \tag{767} \tag{768} \tag{769} \tag{770} \tag{771} \tag{772} \tag{773} \tag{774} \tag{775} \tag{776} \tag{777} \tag{778} \tag{779} \tag{780} \tag{781} \tag{782} \tag{783} \tag{784} \tag{785} \tag{786} \tag{787} \tag{788} \tag{789} \tag{790} \tag{791} \tag{792} \tag{793} \tag{794} \tag{795} \tag{796} \tag{797} \tag{798} \tag{799} \tag{800} \tag{801} \tag{802} \tag{803} \tag{804} \tag{805} \tag{806} \tag{807} \tag{808} \tag{809} \tag{810} \tag{811} \tag{812} \tag{813} \tag{814} \tag{815} \tag{816} \tag{817} \tag{818} \tag{819} \tag{820} \tag{821} \tag{822} \tag{823} \tag{824} \tag{825} \tag{826} \tag{827} \tag{828} \tag{829} \tag{830} \tag{831} \tag{832} \tag{833} \tag{834} \tag{835} \tag{836} \tag{837} \tag{838} \tag{839} \tag{840} \tag{841} \tag{842} \tag{843} \tag{844} \tag{845} \tag{846} \tag{847} \tag{848} \tag{849} \tag{850} \tag{851} \tag{852} \tag{853} \tag{854} \tag{855} \tag{856} \tag{857} \tag{858} \tag{859} \tag{860} \tag{861} \tag{862} \tag{863} \tag{864} \tag{865} \tag{866} \tag{867} \tag{868} \tag{869} \tag{870} \tag{871} \tag{872} \tag{873} \tag{874} \tag{875} \tag{876} \tag{877} \tag{878} \tag{879} \tag{880} \tag{881} \tag{882} \tag{883} \tag{884} \tag{885} \tag{886} \tag{887} \tag{888} \tag{889} \tag{890} \tag{891} \tag{892} \tag{893} \tag{894} \tag{895} \tag{896} \tag{897} \tag{898} \tag{899} \tag{900} \right. \begin{matrix} \tag{901} \tag{902} \tag{903} \tag{904} \tag{905} \tag{906} \tag{907} \tag{908} \tag{909} \tag{910} \tag{911} \tag{912} \tag{913} \tag{914} \tag{915} \tag{916} \tag{917} \tag{918} \tag{919} \tag{920} \tag{921} \tag{922} \tag{923} \tag{924} \tag{925} \tag{926} \tag{927} \tag{928} \tag{929} \tag{930} \tag{931} \tag{932} \tag{933} \tag{934} \tag{935} \tag{936} \tag{937} \tag{938} \tag{939} \tag{940} \tag{941} \tag{942} \tag{943} \tag{944} \tag{945} \tag{946} \tag{947} \tag{948} \tag{949} \tag{950} \tag{951} \tag{952} \tag{953} \tag{954} \tag{955} \tag{956} \tag{957} \tag{958} \tag{959} \tag{960} \tag{961} \tag{962} \tag{963} \tag{964} \tag{965} \tag{966} \tag{967} \tag{968} \tag{969} \tag{970} \tag{971} \tag{972} \tag{973} \tag{974} \tag{975} \tag{976} \tag{977} \tag{978} \tag{979} \tag{980} \tag{981} \tag{982} \tag{983} \tag{984} \tag{985} \tag{986} \tag{987} \tag{988} \tag{989} \tag{990} \tag{991} \tag{992} \tag{993} \tag{994} \tag{995} \tag{996} \tag{997} \tag{998} \tag{999} \tag{1000} \right. \begin{matrix} \tag{1001} \tag{1002} \tag{1003} \tag{1004} \tag{1005} \tag{1006} \tag{1007} \tag{1008} \tag{1009} \tag{1010} \tag{1011} \tag{1012} \tag{1013} \tag{1014} \tag{1015} \tag{1016} \tag{1017} \tag{1018} \tag{1019} \tag{1020} \tag{1021} \tag{1022} \tag{1023} \tag{1024} \tag{1025} \tag{1026} \tag{1027} \tag{1028} \tag{1029} \tag{1030} \tag{1031} \tag{1032} \tag{1033} \tag{1034} \tag{1035} \tag{1036} \tag{1037} \tag{1038} \tag{1039} \tag{1040} \tag{1041} \tag{1042} \tag{1043} \tag{1044} \tag{1045} \tag{1046} \tag{1047} \tag{1048} \tag{1049} \tag{1050} \tag{1051} \tag{1052} \tag{1053} \tag{1054} \tag{1055} \tag{1056} \tag{1057} \tag{1058} \tag{1059} \tag{1060} \tag{1061} \tag{1062} \tag{1063} \tag{1064} \tag{1065} \tag{1066} \tag{1067} \tag{1068} \tag{1069} \tag{1070} \tag{1071} \tag{1072} \tag{1073} \tag{1074} \tag{1075} \tag{1076} \tag{1077} \tag{1078} \tag{1079} \tag{1080} \tag{1081} \tag{1082} \tag{1083} \tag{1084} \tag{1085} \tag{1086} \tag{1087} \tag{1088} \tag{1089} \tag{1090} \tag{1091} \tag{1092} \tag{1093} \tag{1094} \tag{1095} \tag{1096} \tag{1097} \tag{1098} \tag{1099} \tag{1100} \right. \begin{matrix} \tag{1101} \tag{1102} \tag{1103} \tag{1104} \tag{1105} \tag{1106} \tag{1107} \tag{1108} \tag{1109} \tag{1110} \tag{1111} \tag{1112} \tag{1113} \tag{1114} \tag{1115} \tag{1116} \tag{1117} \tag{1118} \tag{1119} \tag{1120} \tag{1121} \tag{1122} \tag{1123} \tag{1124} \tag{1125} \tag{1126} \tag{1127} \tag{1128} \tag{1129} \tag{1130} \tag{1131} \tag{1132} \tag{1133} \tag{1134} \tag{1135} \tag{1136} \tag{1137} \tag{1138} \tag{1139} \tag{1140} \tag{1141} \tag{1142} \tag{1143} \tag{1144} \tag{1145} \tag{1146} \tag{1147} \tag{1148} \tag{1149} \tag{1150} \tag{1151} \tag{1152} \tag{1153} \tag{1154} \tag{1155} \tag{1156} \tag{1157} \tag{1158} \tag{1159} \tag{1160} \tag{1161} \tag{1162} \tag{1163} \tag{1164} \tag{1165} \tag{1166} \tag{1167} \tag{1168} \tag{1169} \tag{1170} \tag{1171} \tag{1172} \tag{1173} \tag{1174} \tag{1175} \tag{1176} \tag{1177} \tag{1178} \tag{1179} \tag{1180} \tag{1181} \tag{1182} \tag{1183} \tag{1184} \tag{1185} \tag{1186} \tag{1187} \tag{1188} \tag{1189} \tag{1190} \tag{1191} \tag{1192} \tag{1193} \tag{1194} \tag{1195} \tag{1196} \tag{1197} \tag{1198} \tag{1199} \tag{1200} \right. \begin{matrix} \tag{1201} \tag{1202} \tag{1203} \tag{1204} \tag{1205} \tag{1206} \tag{1207} \tag{1208} \tag{1209} \tag{1210} \tag{1211} \tag{1212} \tag{1213} \tag{1214} \tag{1215} \tag{1216} \tag{1217} \tag{1218} \tag{1219} \tag{1220} \tag{1221} \tag{1222} \tag{1223} \tag{1224} \tag{1225} \tag{1226} \tag{1227} \tag{1228} \tag{1229} \tag{1230} \tag{1231} \tag{1232} \tag{1233} \tag{1234} \tag{1235} \tag{1236} \tag{1237} \tag{1238} \tag{1239} \tag{1240} \tag{1241} \tag{1242} \tag{1243} \tag{1244} \tag{1245} \tag{1246} \tag{1247} \tag{1248} \tag{1249} \tag{1250} \tag{1251} \tag{1252} \tag{1253} \tag{1254} \tag{1255} \tag{1256} \tag{1257} \tag{1258} \tag{1259} \tag{1260} \tag{1261} \tag{1262} \tag{1263} \tag{1264} \tag{1265} \tag{1266} \tag{1267} \tag{1268} \tag{1269} \tag{1270} \tag{1271} \tag{1272} \tag{1273} \tag{1274} \tag{1275} \tag{1276} \tag{1277} \tag{1278} \tag{1279} \tag{1280} \tag{1281} \tag{1282} \tag{1283} \tag{1284} \tag{1285} \tag{1286} \tag{1287} \tag{1288} \tag{1289} \tag{1290} \tag{1291} \tag{1292} \tag{1293} \tag{1294} \tag{1295} \tag{1296} \tag{1297} \tag{1298} \tag{1299} \tag{1300} \right. \begin{matrix} \tag{1301} \tag{1302} \tag{1303} \tag{1304} \tag{1305} \tag{1306} \tag{1307} \tag{1308} \tag{1309} \tag{1310} \tag{1311} \tag{1312} \tag{1313} \tag{1314} \tag{1315} \tag{1316} \tag{1317} \tag{1318} \tag{1319} \tag{1320} \tag{1321} \tag{1322} \tag{1323} \tag{1324} \tag{1325} \tag{1326} \tag{1327} \tag{1328} \tag{1329} \tag{1330} \tag{1331} \tag{1332} \tag{1333} \tag{1334} \tag{1335} \tag{1336} \tag{1337} \tag{1338} \tag{1339} \tag{1340} \tag{1341} \tag{1342} \tag{1343} \tag{1344} \tag{1345} \tag{1346} \tag{1347} \tag{1348} \tag{1349} \tag{1350} \tag{1351} \tag{1352} \tag{1353} \tag{1354} \tag{1355} \tag{1356} \tag{1357} \tag{1358} \tag{1359} \tag{1360} \tag{1361} \tag{1362} \tag{1363} \tag{1364} \tag{1365} \tag{1366} \tag{1367} \tag{1368} \tag{1369} \tag{1370} \tag{1371} \tag{1372} \tag{1373} \tag{1374} \tag{1375} \tag{1376} \tag{1377} \tag{1378} \tag{1379} \tag{1380} \tag{1381} \tag{1382} \tag{1383} \tag{1384} \tag{1385} \tag{1386} \tag{1387} \tag{1388} \tag{1389} \tag{1390} \tag{1391} \tag{1392} \tag{1393} \tag{1394} \tag{1395} \tag{1396} \tag{1397} \tag{1398} \tag{1399} \tag{1400} \right. \begin{matrix} \tag{1401} \tag{1402} \tag{1403} \tag{1404} \tag{1405} \tag{1406} \tag{1407} \tag{1408} \tag{1409} \tag{1410} \tag{1411} \tag{1412} \tag{1413} \tag{1414} \tag{1415} \tag{1416} \tag{1417} \tag{1418} \tag{1419} \tag{1420} \tag{1421} \tag{1422} \tag{1423} \tag{1424} \tag{1425} \tag{1426} \tag{1427} \tag{1428} \tag{1429} \tag{1430} \tag{1431} \tag{1432} \tag{1433} \tag{1434} \tag{1435} \tag{1436} \tag{1437} \tag{1438} \tag{1439} \tag{1440} \tag{1441} \tag{1442} \tag{1443} \tag{1444} \tag{1445} \tag{1446} \tag{1447} \tag{1448} \tag{1449} \tag{1450} \tag{1451} \tag{1452} \tag{1453} \tag{1454} \tag{1455} \tag{1456} \tag{1457} \tag{1458} \tag{1459} \tag{1460} \tag{1461} \tag{1462} \tag{1463} \tag{1464} \tag{1465} \tag{1466} \tag{1467} \tag{1468} \tag{1469} \tag{1470} \tag{1471} \tag{1472} \tag{1473} \tag{1474} \tag{1475} \tag{1476} \tag{1477} \tag{1478} \tag{1479} \tag{1480} \tag{1481} \tag{1482} \tag{1483} \tag{1484} \tag{1485} \tag{1486} \tag{1487} \tag{1488} \tag{1489} \tag{1490} \tag{1491} \tag{1492} \tag{1493} \tag{1494} \tag{1495} \tag{1496} \tag{1497} \tag{1498} \tag{1499} \tag{1500} \right. \begin{matrix} \tag{1501} \tag{1502} \tag{1503} \tag{1504} \tag{1505} \tag{1506} \tag{1507} \tag{1508} \tag{1509} \tag{1510} \tag{1511} \tag{1512} \tag{1513} \tag{1514} \tag{1515} \tag{1516} \tag{1517} \tag{1518} \tag{1519} \tag{1520} \tag{1521} \tag{1522} \tag{1523} \tag{1524} \tag{1525} \tag{1526} \tag{1527} \tag{1528} \tag{1529} \tag{1530} \tag{1531} \tag{1532} \tag{1533} \tag{1534} \tag{1535} \tag{1536} \tag{1537} \tag{1538} \tag{1539} \tag{1540} \tag{1541} \tag{1542} \tag{1543} \tag{1544} \tag{1545} \tag{1546} \tag{1547} \tag{1548} \tag{1549} \tag{1550} \tag{1551} \tag{1552} \tag{1553} \tag{1554} \tag{1555} \tag{1556} \tag{1557} \tag{1558} \tag{1559} \tag{1560} \tag{1561} \tag{1562} \tag{1563} \tag{1564} \tag{1565} \tag{1566} \tag{1567} \tag{1568} \tag{1569} \tag{1570} \tag{1571} \tag{1572} \tag{1573} \tag{1574} \tag{1575} \tag{1576} \tag{1577} \tag{1578} \tag{1579} \tag{1580} \tag{1581} \tag{1582} \tag{1583} \tag{1584} \tag{1585} \tag{1586} \tag{1587} \tag{1588} \tag{1589} \tag{1590} \tag{1591} \tag{1592} \tag{1593} \tag{1594} \tag{1595} \tag{1596} \tag{1597} \tag{1598} \tag{1599} \tag{1600} \right. \begin{matrix} \tag{1601} \tag{1602} \tag{1603} \tag{1604} \tag{1605} \tag{1606} \tag{1607} \tag{1608} \tag{1609} \tag{1610} \tag{1611} \tag{1612} \tag{1613} \tag{1614} \tag{1615} \tag{1616} \tag{1617} \tag{1618} \tag{1619} \tag{1620} \tag{1621} \tag{1622} \tag{1623} \tag{1624} \tag{1625} \tag{1626} \tag{1627} \tag{1628} \tag{1629} \tag{1630} \tag{1631} \tag{1632} \tag{1633} \tag{1634} \tag{1635} \tag{1636} \tag{1637} \tag{1638} \tag{1639} \tag{1640} \tag{1641} \tag{1642} \tag{1643} \tag{1644} \tag{1645} \tag{1646} \tag{1647} \tag{1648} \tag{1649} \tag{1650} \tag{1651} \tag{1652} \tag{1653} \tag{1654} \tag{1655} \tag{1656} \tag{1657} \tag{1658} \tag{1659} \tag{1660} \tag{1661} \tag{1662} \tag{1663} \tag{1664} \tag{1665} \tag{1666} \tag{1667} \tag{1668} \tag{1669} \tag{1670} \tag{1671} \tag{1672} \tag{1673} \tag{1674} \tag{1675} \tag{1676} \tag{1677} \tag{1678} \tag{1679} \tag{1680} \tag{1681} \tag{1682} \tag{1683} \tag{1684} \tag{1685} \tag{1686} \tag{1687} \tag{1688} \tag{1689} \tag{1690} \tag{1691} \tag{1692} \tag{1693} \tag{1694} \tag{1695} \tag{1696} \tag{1697} \tag{1698} \tag{1699} \tag{1700} \right. \begin{matrix} \tag{1701} \tag{1702} \tag{1703} \tag{1704} \tag{1705} \tag{1706} \tag{1707} \tag{1708} \tag{1709} \tag{1710} \tag{1711} \tag{1712} \tag{1713} \tag{1714} \tag{1715} \tag{1716} \tag{1717} \tag{1718} \tag{1719} \tag{1720} \tag{1721} \tag{1722} \tag{1723} \tag{1724} \tag{1725} \tag{1726} \tag{1727} \tag{1728} \tag{1729} \tag{1730} \tag{1731} \tag{1732} \tag{1733} \tag{1734} \tag{1735} \tag{1736} \tag{1737} \tag{1738} \tag{1739} \tag{1740} \tag{1741} \tag{1742} \tag{1743} \tag{1744} \tag{1745} \tag{1746} \tag{1747} \tag{1748} \tag{1749} \tag{1750} \tag{1751} \tag{1752} \tag{1753} \tag{1754} \tag{1755} \tag{1756} \tag{1757} \tag{1758} \tag{1759} \tag{1760} \tag{1761} \tag{1762} \tag{1763} \tag{1764} \tag{1765} \tag{1766} \tag{1767} \tag{1768} \tag{1769} \tag{1770} \tag{1771} \tag{1772} \tag{1773} \tag{1774} \tag{1775} \tag{1776} \tag{1777} \tag{1778} \tag{1779} \tag{1780} \tag{1781} \tag{1782} \tag{1783} \tag{1784} \tag{1785} \tag{1786} \tag{1787} \tag{1788} \tag{1789} \tag{1790} \tag{1791} \tag{1792} \tag{1793} \tag{1794} \tag{1795} \tag{1796} \tag{1797} \tag{1798} \tag{1799} \tag{1800} \right. \begin{matrix} \tag{1801} \tag{1802} \tag{1803} \tag{1804} \tag{1805} \tag{1806} \tag{1807} \tag{1808} \tag{1809} \tag{1810} \tag{1811} \tag{1812} \tag{1813} \tag{1814} \tag{1815} \tag{1816} \tag{1817} \tag{1818} \tag{1819} \tag{1820} \tag{1821} \tag{1822} \tag{1823} \tag{1824} \tag{1825} \tag{1826} \tag{1827} \tag{1828} \tag{1829} \tag{1830} \tag{1831} \tag{1832} \tag{1833} \tag{1834} \tag{1835} \tag{1836} \tag{1837} \tag{1838} \tag{1839} \tag{1840} \tag{1841} \tag{1842} \tag{1843} \tag{1844} \tag{1845} \tag{1846} \tag{1847} \tag{1848} \tag{1849} \tag{1850} \tag{1851} \tag{1852} \tag{1853} \tag{1854} \tag{1855} \tag{1856} \tag{1857} \tag{1858} \tag{1859} \tag{1860} \tag{1861} \tag{1862} \tag{1863} \tag{1864} \tag{1865} \tag{1866} \tag{1867} \tag{1868} \tag{1869} \tag{1870} \tag{1871} \tag{1872} \tag{1873} \tag{1874} \tag{1875} \tag{1876} \tag{1877} \tag{1878} \tag{1879} \tag{1880} \tag{1881} \tag{1882} \tag{1883} \tag{1884} \tag{1885} \tag{1886} \tag{1887} \tag{1888} \tag{1889} \tag{1890} \tag{1891} \tag{1892} \tag{1893} \tag{1894} \tag{1895} \tag{1896} \tag{1897} \tag{1898} \tag{1899} \tag{1900} \right. \begin{matrix} \tag{1901} \tag{1902} \tag{1903} \tag{1904} \tag{1905} \tag{1906} \tag{1907} \tag{1908} \tag{1909} \tag{1910} \tag{1911} \tag{1912} \tag{1913} \tag{1914} \tag{1915} \tag{1916} \tag{1917} \tag{1918} \tag{1919} \tag{1920} \tag{1921} \tag{1922} \tag{1923} \tag{1924} \tag{1925} \tag{1926} \tag{1927} \tag{1928} \tag{1929} \tag{1930} \tag{1931} \tag{1932} \tag{1933} \tag{1934} \tag{1935} \tag{1936} \tag{1937} \tag{1938} \tag{1939} \tag{1940} \tag{1941} \tag{1942} \tag{1943} \tag{1944} \tag{1945} \tag{1946} \tag{1947} \tag{1948} \tag{1949} \tag{1950} \tag{1951} \tag{1952} \tag{1953} \tag{1954} \tag{1955} \tag{1956} \tag{1957} \tag{1958} \tag{1959} \tag{1960} \tag{1961} \tag{1962} \tag{1963} \tag{1964} \tag{1965} \tag{1966} \tag{1967} \tag{1968} \tag{1969} \tag{1970} \tag{1971} \tag{1972} \tag{1973} \tag{1974} \tag{1975} \tag{1976} \tag{1977} \tag{1978} \tag{1979} \tag{1980} \tag{1981} \tag{1982} \tag{1983} \tag{1984} \tag{1985} \tag{1986} \tag{1987} \tag{1988} \tag{1989} \tag{1990} \tag{1991} \tag{1992} \tag{1993} \tag{1994} \tag{1995} \tag{1996} \tag{1997} \tag{1998} \tag{1999} \tag{2000} \right. \begin{matrix} \tag{2001} \tag{2002} \tag{2003} \tag{2004} \tag{2005} \tag{2006} \tag{2007} \tag{2008} \tag{2009} \tag{2010} \tag{2011} \tag{2012} \tag{2013} \tag{2014} \tag{2015} \tag{2016} \tag{2017} \tag{2018} \tag{2019} \tag{2020} \tag{2021} \tag{2022} \tag{2023} \tag{2024} \tag{2025} \tag{2026} \tag{2027} \tag{2028} \tag{2029} \tag{2030} \tag{2031} \tag{2032} \tag{2033} \tag{2034} \tag{2035} \tag{2036} \tag{2037} \tag{2038} \tag{2039} \tag{2040} \tag{2041} \tag{2042} \tag{2043} \tag{2044} \tag{2045} \tag{2046} \tag{2047} \tag{2048} \tag{2049} \tag{2050} \tag{2051} \tag{2052} \tag{2053} \tag{2054} \tag{2055} \tag{2056} \tag{2057} \tag{2058} \tag{2059} \tag{2060} \tag{2061} \tag{2062} \tag{2063} \tag{2064} \tag{2065} \tag{2066} \tag{2067} \tag{2068} \tag{2069} \tag{2070} \tag{2071} \tag{2072} \tag{2073} \tag{2074} \tag{2075} \tag{2076} \tag{2077} \tag{2078} \tag{2079} \tag{2080} \tag{2081} \tag{2082} \tag{2083} \tag{2084} \tag{2085} \tag{2086} \tag{2087} \tag{2088} \tag{2089} \tag{2090} \tag{2091} \tag{2092} \tag{2093} \tag{2094} \tag{2095} \tag{2096} \tag{2097} \tag{2098} \tag{2099} \tag{2100} \right. \begin{matrix} \tag{2101} \tag{2102} \tag{2103} \tag{2104} \tag{2105} \tag{2106} \tag{2107} \tag{2108} \tag{2109} \tag{2110} \tag{2111} \tag{2112} \tag{2113} \tag{2114} \tag{2115} \tag{2116} \tag{2117} \tag{2118} \tag{2119} \tag{2120} \tag{2121} \tag{2122} \tag{2123} \tag{2124} \tag{2125} \tag{2126} \tag{2127} \tag{2128} \tag{2129} \tag{2130} \tag{2131} \tag{2132} \tag{2133} \tag{2134} \tag{2135} \tag{2136} \tag{2137} \tag{2138} \tag{2139} \tag{2140} \tag{2141} \tag{2142} \tag{2143} \tag{2144} \tag{2145} \tag{2146} \tag{2147} \tag{2148} \tag{2149} \tag{2150} \tag{2151} \tag{2152} \tag{2153} \tag{2154} \tag{2155} \tag{2156} \tag{2157} \tag{2158} \tag{2159} \tag{2160} \tag{2161} \tag{2162} \tag{2163} \tag{2164} \tag{2165} \tag{2166} \tag{2167} \tag{2168} \tag{2169} \tag{2170} \tag{2171} \tag{2172} \tag{2173} \tag{2174} \tag{2175} \tag{2176} \tag{2177} \tag{2178} \tag{2179} \tag{2180} \tag{2181} \tag{2182} \tag{2183} \tag{2184} \tag{2185} \tag{2186} \tag{2187} \tag{2188} \tag{2189} \tag{2190} \tag{2191} \tag{2192} \tag{2193} \tag{2194} \tag{2195} \tag{2196} \tag{2197} \tag{2198} \tag{2199} \tag{2200} \right. \begin{matrix} \tag{2201} \tag{2202} \tag{2203} \tag{2204} \tag{2205} \tag{2206} \tag{2207} \tag{2208} \tag{2209} \tag{2210} \tag{2211} \tag{2212} \tag{2213} \tag{2214} \tag{2215} \tag{2216} \tag{2217} \tag{2218} \tag{2219} \tag{2220} \tag{2221} \tag{2222} \tag{2223} \tag{2224} \tag{2225} \tag{2226} \tag{2227} \tag{2228} \tag{2229} \tag{2230} \tag{2231} \tag{2232} \tag{2233} \tag{2234} \tag{2235} \tag{2236} \tag{2237} \tag{2238} \tag{2239} \tag{2240} \tag{2241} \tag{2242} \tag{2243} \tag{2244} \tag{2245} \tag{2246} \tag{2247} \tag{2248} \tag{2249} \tag{2250} \tag{2251} \tag{2252} \tag{2253} \tag{2254} \tag{2255} \tag{2256} \tag{2257} \tag{2258} \tag{2259} \tag{2260} \tag{2261} \tag{2262} \tag{2263} \tag{2264} \tag{2265} \tag{2266} \tag{2267} \tag{2268} \tag{2269} \tag{2270} \tag{2271} \tag{2272} \tag{2273} \tag{2274} \tag{2275} \tag{2276} \tag{2277} \tag{2278} \tag{2279} \tag{2280} \tag{2281} \tag{2282} \tag{2283} \tag{2284} \tag{2285} \tag{2286} \tag{2287} \tag{2288} \tag{2289} \tag{2290} \tag{2291} \tag{2292} \tag{2293} \tag{2294} \tag{2295} \tag{2296} \tag{2297} \tag{2298} \tag{2299} \tag{2300} \right. \begin{matrix} \tag{2301} \tag{2302} \tag{2303} \tag{2304} \tag{2305} \tag{2306} \tag{2307} \tag{2308} \tag{2309} \tag{2310} \tag{2311} \tag{2312} \tag{2313} \tag{2314} \tag{2315} \tag{2316} \tag{2317} \tag{2318} \tag{2319} \tag{2320} \tag{2321} \tag{2322} \tag{2323} \tag{2324} \tag{2325} \tag{2326} \tag{2327} \tag{2328} \tag{2329} \tag{2330} \tag{2331} \tag{2332} \tag{2333} \tag{2334} \tag{2335} \tag{2336} \tag{2337} \tag{2338} \tag{2339} \tag{2340} \tag{2341} \tag{2342} \tag{2343} \tag{2344} \tag{2345} \tag{2346} \tag{2347} \tag{2348} \tag{2349} \tag{2350} \tag{2351} \tag{2352} \tag{2353} \tag{2354} \tag{2355} \tag{2356} \tag{2357} \tag{2358} \tag{2359} \tag{2360} \tag{2361} \tag{2362} \tag{2363} \tag{2364} \tag{2365} \tag{2366} \tag{2367} \tag{2368} \tag{2369} \tag{2370} \tag{2371} \tag{2372} \tag{2373} \tag{2374} \tag{2375} \tag{2376} \tag{2377} \tag{2378} \tag{2379} \tag{2380} \tag{2381} \tag{2382} \tag{2383} \tag{2384} \tag{2385} \tag{2386} \tag{2387} \tag{2388} \tag{2389} \tag{2390} \tag{2391} \tag{2392} \tag{2393} \tag{2394} \tag{2395} \tag{2396} \tag{2397} \tag{2398} \tag{2399} \tag{2400} \right. \begin{matrix} \tag{2401} \tag{2402} \tag{2403} \tag{2404} \tag{2405} \tag{2406} \tag{2407} \tag{2408} \tag{2409} \tag{2410} \tag{2411} \tag{2412} \tag{2413} \tag{2414} \tag{2415} \tag{2416} \tag{2417} \tag{2418} \tag{2419} \tag{2420} \tag{2421} \tag{2422} \tag{2423} \tag{2424} \tag{2425} \tag{2426} \tag{2427} \tag{2428} \tag{2429} \tag{2430} \tag{2431} \tag{2432} \tag{2433} \tag{2434} \tag{2435} \tag{2436} \tag{2437} \tag{2438} \tag{2439} \tag{2440} \tag{2441} \tag{2442} \tag{2443} \tag{2444} \tag{2445} \tag{2446} \tag{2447} \tag{2448} \tag{2449} \tag{2450} \tag{2451} \tag{2452} \tag{2453} \tag{2454} \tag{2455} \tag{2456} \tag{2457} \tag{2458} \tag{2459} \tag{2460} \tag{2461} \tag{2462} \tag{2463} \tag{2464} \tag{2465} \tag{2466} \tag{2467} \tag{2468} \tag{2469} \tag{2470} \tag{2471} \tag{2472} \tag{2473} \tag{2474} \tag{2475} \tag{2476} \tag{2477} \tag{2478} \tag{2479} \tag{2480} \tag{2481} \tag{2482} \tag{2483} \tag{2484} \tag{2485} \tag{2486} \tag{2487} \tag{2488} \tag{2489} \tag{2490} \tag{2491} \tag{2492} \tag{2493} \tag{2494} \tag{2495} \tag{2496} \tag{2497} \tag{2498} \tag{2499} \tag{2500} \right. \begin{matrix} \tag{2501} \tag{2502} \tag{2503} \tag{2504} \tag{2505} \tag{2506} \tag{2507} \tag{2508} \tag{2509} \tag{2510} \tag{2511} \tag{2512} \tag{2513} \tag{2514} \tag{2515} \tag{2516} \tag{2517} \tag{2518} \tag{2519} \tag{2520} \tag{2521} \tag{2522} \tag{2523} \tag{2524} \tag{2525} \tag{2526} \tag{2527} \tag{2528} \tag{2529} \tag{2530} \tag{2531} \tag{2532} \tag{2533} \tag{2534} \tag{2535} \tag{2536} \tag{2537} \tag{2538} \tag{2539} \tag{2540} \tag{2541} \tag{2542} \tag{2543} \tag{2544} \tag{2545} \tag{2546} \tag{2547} \tag{2548} \tag{2549} \tag{2550} \tag{2551} \tag{2552} \tag{2553} \tag{2554} \tag{2555} \tag{2556} \tag{2557} \tag{2558} \tag{2559} \tag{2560} \tag{2561} \tag{2562} \tag{2563} \tag{2564} \tag{2565} \tag{2566} \tag{2567} \tag{2568} \tag{2569} \tag{2570} \tag{2571} \tag{2572} \tag{2573} \tag{2574} \tag{2575} \tag{2576} \tag{2577} \tag{2578} \tag{2579} \tag{2580} \tag{2581} \tag{2582} \tag{2583} \tag{2584} \tag{2585} \tag{2586} \tag{2587} \tag{2588} \tag{2589} \tag{2590} \tag{2591} \tag{2592} \tag{2593} \tag{2594} \tag{2595} \tag{2596} \tag{2597} \tag{2598} \tag{2599} \tag{2600} \right. \begin{matrix} \tag{2601} \tag{2602} \tag{2603} \tag{2604} \tag{2605} \tag{2606} \tag{2607} \tag{2608} \tag{2609} \tag{2610} \tag{2611} \tag{2612} \tag{2613} \tag{2614} \tag{2615} \tag{2616} \tag{2617} \tag{2618} \tag{2619} \tag{2620} \tag{2621} \tag{2622} \tag{2623} \tag{2624} \tag{2625} \tag{2626} \tag{2627} \tag{2628} \tag{2629} \tag{2630} \tag{2631} \tag{2632} \tag{2633} \tag{2634} \tag{2635} \tag{2636} \tag{2637} \tag{2638} \tag{2639} \tag{2640} \tag{2641} \tag{2642} \tag{2643} \tag{2644} \tag{2645} \tag{2646} \tag{2647} \tag{2648} \tag{2649} \tag{2650} \tag{2651} \tag{2652} \tag{2653} \tag{2654} \tag{2655} \tag{2656} \tag{2657} \tag{2658} \tag{2659} \tag{2660} \tag{2661} \tag{2662} \tag{2663} \tag{2664} \tag{2665} \tag{2666} \tag{2667} \tag{2668} \tag{2669} \tag{2670} \tag{2671} \tag{2672} \tag{2673} \tag{2674} \tag{2675} \tag{2676} \tag{2677} \tag{2678} \tag{2679} \tag{2680} \tag{2681} \tag{2682} \tag{2683} \tag{2684} \tag{2685} \tag{2686} \tag{2687} \tag{2688} \tag{2689} \tag{2690} \tag{2691} \tag{2692} \tag{2693} \tag{2694} \tag{2695} \tag{2696} \tag{2697} \tag{2698} \tag{2699} \tag{2700} \right. \begin{matrix} \tag{2701} \tag{2702} \tag{2703} \tag{2704} \tag{2705} \tag{2706} \tag{2707} \tag{2708} \tag{2709} \tag{2710} \tag{2711} \tag{2712} \tag{2713} \tag{2714} \tag{2715} \tag{2716} \tag{2717} \tag{2718} \tag{2719} \tag{2720} \tag{2721} \tag{2722} \tag{2723} \tag{2724} \tag{2725} \tag{2726} \tag{2727} \tag{2728} \tag{2729} \tag{2730} \tag{2731} \tag{2732} \tag{2733} \tag{2734} \tag{2735} \tag{2736} \tag{2737} \tag{2738} \tag{2739} \tag{2740} \tag{2741} \tag{2742} \tag{2743} \tag{2744} \tag{2745} \tag{2746} \tag{2747} \tag{2748} \tag{2749} \tag{2750} \tag{2751} \tag{2752} \tag{2753} \tag{2754} \tag{2755} \tag{2756} \tag{2757} \tag{2758} \tag{2759} \tag{2760} \tag{2761} \tag{2762} \tag{2763} \tag{2764} \tag{2765} \tag{2766} \tag{2767} \tag{2768} \tag{2769} \tag{2770} \tag{2771} \tag{2772} \tag{2773} \tag{2774} \tag{2775} \tag{2776} \tag{2777} \tag{2778} \tag{2779} \tag{2780} \tag{2781} \tag{2782} \tag{2783} \tag{2784} \tag{2785} \tag{2786} \tag{2787} \tag{2788} \tag{2789} \tag{2790} \tag{2791} \tag{2792} \tag{2793} \tag{2794} \tag{2795} \tag{2796} \tag{2797} \tag{2798} \tag{2799} \tag{2800} \right. \begin{matrix} \tag{2801} \tag{2802} \tag{2803} \tag{2804} \tag{2805} \tag{2806} \tag{2807} \tag{2808} \tag{2809} \tag{2810} \tag{2811} \tag{2812} \tag{2813} \tag{2814} \tag{2815} \tag{2816} \tag{2817} \tag{2818} \tag{2819} \tag{2820} \tag{2821} \tag{2822} \tag{2823} \tag{2824} \tag{2825} \tag{2826} \tag{2827} \tag{2828} \tag{2829} \tag{2830} \tag{2831} \tag{2832} \tag{2833} \tag{2834} \tag{2835} \tag{2836} \tag{2837} \tag{2838} \tag{2839} \tag{2840} \tag{2841} \tag{2842} \tag{2843} \tag{2844} \tag{2845} \tag{2846} \tag{2847} \tag{2848} \tag{2849} \tag{2850} \tag{2851} \tag{2852} \tag{2853} \tag{2854} \tag{2855} \tag{2856} \tag{2857} \tag{2858} \tag{2859} \tag{2860} \tag{2861} \tag{2862} \tag{2863} \tag{2864} \tag{2865} \tag{2866} \tag{2867} \tag{2868} \tag{2869} \tag{2870} \tag{2871} \tag{2872} \tag{2873} \tag{2874} \tag{2875} \tag{2876} \tag{2877} \tag{2878} \tag{2879} \tag{2880} \tag{2881} \tag{2882} \tag{2883} \tag{2884} \tag{2885} \tag{2886} \tag{2887} \tag{2888} \tag{2889} \tag{2890} \tag{2891} \tag{2892} \tag{2893} \tag{2894} \tag{2895} \tag{2896} \tag{2897} \tag{2898} \tag{2899} \tag{2900} \right. \begin{matrix} \tag{2901} \tag{2902} \tag{2903} \tag{2904} \tag{2905} \tag{2906} \tag{2907} \tag{2908} \tag{2909} \tag{2910} \tag{2911} \tag{2912} \tag{2913} \tag{2914} \tag{2915} \tag{2916} \tag{2917} \tag{2918} \tag{2919} \tag{2920} \tag{2921} \tag{2922} \tag{2923} \tag{2924} \tag{2925} \tag{2926} \tag{2927} \tag{2928} \tag{2929} \tag{2930} \tag{2931} \tag{2932} \tag{2933} \tag{2934} \tag{2935} \tag{2936} \tag{2937} \tag{2938} \tag{2939} \tag{2940} \tag{2941} \tag{2942} \tag{2943} \tag{2944} \tag{2945} \tag{2946} \tag{2947} \tag{2948} \tag{2949} \tag{2950} \tag{2951} \tag{2952} \tag{2953} \tag{2954} \tag{2955} \tag{2956} \tag{2957} \tag{2958} \tag{2959} \tag{2960} \tag{2961} \tag{2962} \tag{2963} \tag{2964} \tag{2965} \tag{2966} \tag{2967} \tag{2968} \tag{2969} \tag{2970} \tag{2971} \tag{2972} \tag{2973} \tag{2974} \tag{2975} \tag{2976} \tag{2977} \tag{2978} \tag{2979} \tag{2980} \tag{2981} \tag{2982} \tag{2983} \tag{2984} \tag{2985} \tag{2986} \tag{2987} \tag{2988} \tag{2989} \tag{2990} \tag{2991} \tag{2992} \tag{2993} \tag{2994} \tag{2995} \tag{2996} \tag{2997} \tag{2998} \tag{2999} \tag{3000} \right. \begin{matrix} \tag{3001} \tag{3002} \tag{3003} \tag{3004} \tag{3005} \tag{3006} \tag{3007} \tag{3008} \tag{3009} \tag{3010} \tag{3011} \tag{3012} \tag{3013} \tag{3014} \tag{3015} \tag{3016} \tag{3017} \tag{3018} \tag{3019} \tag{3020} \tag{3021} \tag{3022} \tag{3023} \tag{3024} \tag{3025} \tag{3026} \tag{3027} \tag{3028} \tag{3029} \tag{3030} \tag{3031} \tag{3032} \tag{3033} \tag{3034} \tag{3035} \tag{3036} \tag{3037} \tag{3038} \tag{3039} \tag{3040} \tag{3041} \tag{3042} \tag{3043} \tag{3044} \tag{3045} \tag{3046} \tag{3047} \tag{3048} \tag{3049} \tag{3050} \tag{3051} \tag{3052} \tag{3053} \tag{3054} \tag{3055} \tag{3056} \tag{3057} \tag{3058} \tag{3059} \tag{3060} \tag{3061} \tag{3062} \tag{3063} \tag{3064} \tag{3065} \tag{3066} \tag{3067} \tag{3068} \tag{3069} \tag{3070} \tag{3071} \tag{3072} \tag{3073} \tag{3074} \tag{3075} \tag{3076} \tag{3077} \tag{3078} \tag{3079} \tag{3080} \tag{3081} \tag{3082} \tag{3083} \tag{3084} \tag{3085} \tag{3086} \tag{3087} \tag{3088} \tag{3089} \tag{3090} \tag{3091} \tag{3092} \tag{3093} \tag{3094} \tag{3095} \tag{3096} \tag{3097} \tag{3098} \tag{3099} \tag{3100} \right. \begin{matrix} \tag{3101} \tag{3102} \tag{3103} \tag{3104} \tag{3105} \tag{3106} \tag{3107} \tag{3108} \tag{3109} \tag{3110} \tag{3111} \tag{3112} \tag{3113} \tag{3114} \tag{3115} \tag{3116} \tag{3117} \tag{3118} \tag{3119} \tag{3120} \tag{3121} \tag{3122} \tag{3123} \tag{3124} \tag{3125} \tag{3126} \tag{3127} \tag{3128} \tag{3129} \tag{3130} \tag{3131} \tag{3132} \tag{3133} \tag{3134} \tag{3135} \tag{3136} \tag{3137} \tag{3138} \tag{3139} \tag{3140} \tag{3141} \tag{3142} \tag{3143} \tag{3144} \tag{3145} \tag{3146} \tag{3147} \tag{3148} \tag{3149} \tag{3150} \tag{3151} \tag{3152} \tag{3153} \tag{3154} \tag{3155} \tag{3156} \tag{3157} \tag{3158} \tag{3159} \tag{3160} \tag{3161} \tag{3162} \tag{3163} \tag{3164} \tag{3165} \tag{3166} \tag{3167} \tag{3168} \tag{3169} \tag{3170} \tag{3171} \tag{3172} \tag{3173} \tag{3174} \tag{3175} \tag{3176} \tag{3177} \tag{3178} \tag{3179} \tag{3180} \tag{3181} \tag{3182} \tag{3183} \tag{3184} \tag{3185} \tag{3186} \tag{3187} \tag{3188} \tag{3189} \tag{3190} \tag{3191} \tag{3192} \tag{3193} \tag{3194} \tag{3195} \tag{3196} \tag{3197} \tag{3198} \tag{3199} \tag{3200} \right. \begin{matrix} \tag{3201} \tag{3202} \tag{3203} \tag{3204} \tag{3205} \tag{3206} \tag{3207} \tag{3208} \tag{3209} \tag{3210} \tag{3211} \tag{3212} \tag{3213} \tag{3214} \tag{3215} \tag{3216} \tag{3217} \tag{3218} \tag{3219} \tag{3220} \tag{3221} \tag{3222} \tag{3223} \tag{3224} \tag{3225} \tag{3226} \tag{3227} \tag{3228} \tag{3229} \tag{3230} \tag{3231} \tag{3232} \tag{3233} \tag{3234} \tag{3235} \tag{3236} \tag{3237} \tag{3238} \tag{3239} \tag{3240} \tag{3241} \tag{3242} \tag{3243} \tag{3244} \tag{3245} \tag{3246} \tag{3247} \tag{3248} \tag{3249} \tag{3250} \tag{3251} \tag{3252} \tag{3253} \tag{3254} \tag{3255} \tag{3256} \tag{3257} \tag{3258} \tag{3259} \tag{3260} \tag{3261} \tag{3262} \tag{3263} \tag{3264} \tag{3265} \tag{3266} \tag{3267} \tag{3268} \tag{3269} \tag{3270} \tag{3271} \tag{3272} \tag{3273} \tag{3274} \tag{3275} \tag{3276} \tag{3277} \tag{3278} \tag{3279} \tag{3280} \tag{3281} \tag{3282} \tag{3283} \tag{3284} \tag{3285} \tag{3286} \tag{3287} \tag{3288} \tag{3289} \tag{3290} \tag{3291} \tag{3292} \tag{3293} \tag{3294} \tag{3295} \tag{3296} \tag{3297} \tag{3298} \tag{3299} \tag{3300} \right. \begin{matrix} \tag{3301} \tag{3302} \tag{3303} \tag{3304} \tag{3305} \tag{3306} \tag{3307} \tag{3308} \tag{3309} \tag{3310} \tag{3311} \tag{3312} \tag{3313} \tag{3314} \tag{3315} \tag{3316} \tag{3317} \tag{3318} \tag{3319} \tag{3320} \tag{3321} \tag{3322} \tag{3323} \tag{3324} \tag{3325} \tag{3326} \tag{3327} \tag{3328} \tag{3329} \tag{3330} \tag{3331} \tag{3332} \tag{3333} \tag{3334} \tag{3335} \tag{3336} \tag{3337} \tag{3338} \tag{3339} \tag{3340} \tag{3341} \tag{3342} \tag{3343} \tag{3344} \tag{3345} \tag{3346} \tag{3347} \tag{3348} \tag{3349} \tag{3350} \tag{3351} \tag{3352} \tag{3353} \tag{3354} \tag{3355} \tag{3356} \tag{3357} \tag{3358} \tag{3359} \tag{3360} \tag{3361} \tag{3362} \tag{3363} \tag{3364} \tag{3365} \tag{3366} \tag{3367} \tag{3368} \tag{3369} \tag{3370} \tag{3371} \tag{3372} \tag{3373} \tag{3374} \tag{3375} \tag{3376} \tag{3377} \tag{3378} \tag{3379} \tag{3380} \tag{3381} \tag{3382} \tag{3383} \tag{3384} \tag{3385} \tag{3386} \tag{3387} \tag{3388} \tag{3389} \tag{3390} \tag{3391} \tag{3392} \tag{3393} \tag{3394} \tag{3395} \tag{3396} \tag{3397} \tag{3398} \tag{3399} \tag{3400} \right. \begin{matrix} \tag{3401} \tag{3402} \tag{3403} \tag{3404} \tag{3405} \tag{3406} \tag{3407} \tag{3408} \tag{3409} \tag{3410} \tag{3411} \tag{3412} \tag{3413} \tag{3414} \tag{3415} \tag{3416} \tag{3417} \tag{3418} \tag{3419} \tag{3420} \tag{3421} \tag{3422} \tag{3423} \tag{3424} \tag{3425} \tag{3426} \tag{3427} \tag{3428} \tag{3429} \tag{3430} \tag{3431} \tag{3432} \tag{3433} \tag{3434} \tag{3435} \tag{3436} \tag{3437} \tag{3438} \tag{3439} \tag{3440} \tag{3441} \tag{3442} \tag{3443} \tag{3444} \tag{3445} \tag{3446} \tag{3447} \tag{3448} \tag{3449} \tag{3450} \tag{3451} \tag{3452} \tag{3453} \tag{3454} \tag{3455} \tag{3456} \tag{3457} \tag{3458} \tag{3459} \tag{3460} \tag{3461} \tag{3462} \tag{3463} \tag{3464} \tag{3465} \tag{3466} \tag{3467} \tag{3468} \tag{3469} \tag{3470} \tag{3471} \tag{3472} \tag{3473} \tag{3474} \tag{3475} \tag{3476} \tag{3477} \tag{3478} \tag{3479} \tag{3480} \tag{3481} \tag{3482} \tag{3483} \tag{3484} \tag{3485} \tag{3486} \tag{3487} \tag{3488} \tag{3489} \tag{3490} \tag{3491} \tag{3492} \tag{3493} \tag{3494} \tag{3495} \tag{3496} \tag{3497} \tag{3498} \tag{3499} \tag{3500} \right. \begin{matrix} \tag{3501} \tag{3502} \tag{3503} \tag{3504} \tag{3505} \tag{3506} \tag{3507} \tag{3508} \tag{3509} \tag{3510} \tag{3511} \tag{3512} \tag{3513} \tag{3514} \tag{3515} \tag{3516} \tag{3517} \tag{3518} \tag{3519} \tag{3520} \tag{3521} \tag{3522} \tag{3523} \tag{3524} \tag{3525} \tag{3526} \tag{3527} \tag{3528} \tag{3529} \tag{3530} \tag{3531} \tag{3532} \tag{3533} \tag{3534} \tag{3535} \tag{3536} \tag{3537} \tag{3538} \tag{3539} \tag{3540} \tag{3541} \tag{3542} \tag{3543} \tag{3544} \tag{3545} \tag{3546} \tag{3547} \tag{3548} \tag{3549} \tag{3550} \tag{3551} \tag{3552} \tag{3553} \tag{3554} \tag{3555} \tag{3556} \tag{3557} \tag{3558} \tag{3559} \tag{3560} \tag{3561} \tag{3562} \tag{3563} \tag{3564} \tag{3565} \tag{3566} \tag{3567} \tag{3568} \tag{3569} \tag{3570} \tag{3571} \tag{3572} \tag{3573} \tag{3574} \tag{3575} \tag{3576} \tag{3577} \tag{3578} \tag{3579} \tag{3580} \tag{3581} \tag{3582} \tag{3583} \tag{3584} \tag{3585} \tag{3586} \tag{3587} \tag{3588} \tag{3589} \tag{3590} \tag{3591} \tag{3592} \tag{3593} \tag{3594} \tag{3595} \tag{3596} \tag{3597} \tag{3598} \tag{3599} \tag{3600} \right. \begin{matrix} \tag{3601} \tag{3602} \tag{3603} \tag{3604} \tag{3605} \tag{3606} \tag{3607} \tag{3608} \tag{3609} \tag{3610} \tag{3611} \tag{3612} \tag{3613} \tag{3614} \tag{3615} \tag{3616} \tag{3617} \tag{3618} \tag{3619} \tag{3620} \tag{3621} \tag{3622} \tag{3623} \tag{3624} \tag{3625} \tag{3626} \tag{3627} \tag{3628} \tag{3629} \tag{3630} \tag{3631} \tag{3632} \tag{3633} \tag{3634} \tag{3635} \tag{3636} \tag{3637} \tag{3638} \tag{3639} \tag{3640} \tag{3641} \tag{3642} \tag{3643} \tag{3644} \tag{3645} \tag{3646} \tag{3647} \tag{3648} \tag{3649} \tag{3650} \tag{3651} \tag{3652} \tag{3653} \tag{3654} \tag{3655} \tag{3656} \tag{3657} \tag{3658} \tag{3659} \tag{3660} \tag{3661} \tag{3662} \tag{3663} \tag{3664} \tag{3665} \tag{3666} \tag{3667} \tag{3668} \tag{3669} \tag{3670} \tag{3671} \tag{3672} \tag{3673} \tag{3674} \tag{3675} \tag{3676} \tag{3677} \tag{3678} \tag{3679} \tag{3680} \tag{3681} \tag{3682} \tag{3683} \tag{3684} \tag{3685} \tag{3686} \tag{3687} \tag{3688} \tag{3689} \tag{3690} \tag{3691} \tag{3692} \tag{3693} \tag{3694} \tag{3695} \tag{3696} \tag{3697} \tag{3698} \tag{3699} \tag{3700} \right. \begin{matrix} \tag{3701} \tag{3702} \tag{3703} \tag{3704} \tag{3705} \tag{3706} \tag{3707} \tag{3708} \tag{3709} \tag{3710} \tag{3711} \tag{3712} \tag{3713} \tag{3714} \tag{3715} \tag{3716} \tag{3717} \tag{3718} \tag{3719} \tag{3720} \tag{3721} \tag{3722} \tag{3723} \tag{3724} \tag{3725} \tag{3726} \tag{3727} \tag{3728} \tag{3729} \tag{3730} \tag{3731} \tag{3732} \tag{3733} \tag{3734} \tag{3735} \tag{3736} \tag{3737} \tag{3738} \tag{3739} \tag{3740} \tag{3741} \tag{3742} \tag{3743} \tag{3744} \tag{3745} \tag{3746} \tag{3747} \tag{3748} \tag{3749} \tag{3750} \tag{3751} \tag{3752} \tag{3753} \tag{3754} \tag{3755} \tag{3756} \tag{3757} \tag{3758} \tag{3759} \tag{3760} \tag{3761} \tag{3762} \tag{3763} \tag{3764} \tag{3765} \tag{3766} \tag{3767} \tag{3768} \tag{3769} \tag{3770} \tag{3771} \tag{3772} \tag{3773} \tag{3774} \tag{3775} \tag{3776} \tag{3777} \tag{3778} \tag{3779} \tag{3780} \tag{3781} \tag{3782} \tag{3783} \tag{3784} \tag{3785} \tag{3786} \tag{3787} \tag{3788} \tag{3789} \tag{3790} \tag{3791} \tag{3792} \tag{3793} \tag{3794} \tag{3795} \tag{3796} \tag{3797} \tag{3798} \tag{3799} \tag{3800} \right. \begin{matrix} \tag{3801} \tag{3802} \tag{3803} \tag{3804} \tag{3805} \tag{3806} \tag{3807} \tag{3808} \tag{3809} \tag{3810} \tag{3811} \tag{3812} \tag{3813} \tag{3814} \tag{3815} \tag{3816} \tag{3817} \tag{3818} \tag{3819} \tag{3820} \tag{3821} \tag{3822} \tag{3823} \tag{3824} \tag{3825} \tag{3826} \tag{3827} \tag{3828} \tag{3829} \tag{3830} \tag{3831} \tag{3832} \tag{3833} \tag{3834} \tag{3835} \tag{3836} \tag{3837} \tag{3838} \tag{3839} \tag{3840} \tag{3841} \tag{3842} \tag{3843} \tag{3844} \tag{3845} \tag{3846} \tag{3847} \tag{3848} \tag{3849} \tag{3850} \tag{3851} \tag{3852} \tag{3853} \tag{3854} \tag{3855} \tag{3856} \tag{3857} \tag{3858} \tag{3859} \tag{3860} \tag{3861} \tag{3862} \tag{3863} \tag{3864} \tag{3865} \tag{3866} \tag{3867} \tag{3868} \tag{3869} \tag{3870} \tag{3871} \tag{3872} \tag{3873} \tag{3874} \tag{3875} \tag{3876} \tag{3877} \tag{3878} \tag{3879} \tag{3880} \tag{3881} \tag{3882} \tag{3883} \tag{3884} \tag{3885} \tag{3886} \tag{3887} \tag{3888} \tag{3889} \tag{3890} \tag{3891} \tag{3892} \tag{3893} \tag{3894} \tag{3895} \tag{3896} \tag{3897} \tag{3898} \tag{3899} \tag{3900} \right. \begin{matrix} \tag{3901} \tag{3902} \tag{3903} \tag{3904} \tag{3905} \tag{3906} \tag{3907} \tag{3908} \tag{3909} \tag{3910} \tag{3911} \tag{3912} \tag{3913} \tag{3914} \tag{3915} \tag{3916} \tag{3917} \tag{3918} \tag{3919} \tag{3920} \tag{3921} \tag{3922} \tag{3923} \tag{3924} \tag{3925} \tag{3926} \tag{3927} \tag{3928} \tag{3929} \tag{3930} \tag{3931} \tag{3932} \tag{3933} \tag{3934} \tag{3935} \tag{3936} \tag{3937} \tag{3938} \tag{3939} \tag{3940} \tag{3941} \tag{3942} \tag{3943} \tag{3944} \tag{3945} \tag{3946} \tag{3947} \tag{3948} \tag{3949} \tag{3950} \tag{3951} \tag{3952} \tag{3953} \tag{3954} \tag{3955} \tag{3956} \tag{3957} \tag{3958} \tag{3959} \tag{3960} \tag{3961} \tag{3962} \tag{3963} \tag{3964} \tag{3965} \tag{3966} \tag{3967} \tag{3968} \tag{3969} \tag{3970} \tag{3971} \tag{3972} \tag{3973} \tag{3974} \tag{3975} \tag{3976} \tag{3977} \tag{3978} \tag{3979} \tag{3980} \tag{3981} \tag{3982} \tag{3983} \tag{3984} \tag{3985} \tag{3986} \tag{3987} \tag{3988} \tag{3989} \tag{3990} \tag{3991} \tag{3992} \tag{3993} \tag{3994} \tag{3995} \tag{3996} \tag{3997} \tag{3998} \tag{3999} \tag{4000} \right. f(x_1) \begin{matrix} \tag{4001} \tag{4002} \tag{4003} \tag{4004} \tag{4005} \tag{4006} \tag{4007} \tag{4008} \tag{4009} \tag{4010} \tag{4011} \tag{4012} \tag{4013} \tag{4014} \tag{4015} \tag{4016} \tag{4017} \tag{4018} \tag{4019} \tag{4020} \tag{4021} \tag{4022} \tag{4023} \tag{4024} \tag{4025} \tag{4026} \tag{4027} \tag{4028} \tag{4029} \tag{4030} \tag{4031} \tag{4032} \tag{4033} \tag{4034} \tag{4035} \tag{4036} \tag{4037} \tag{4038} \tag{4039} \tag{4040} \tag{4041} \tag{4042} \tag{4043} \tag{4044} \tag{4045} \tag{4046} \tag{4047} \tag{4048} \tag{4049} \tag{4050} \tag{4051} \tag{4052} \tag{4053} \tag{4054} \tag{4055} \tag{4056} \tag{4057} \tag{4058} \tag{4059} \tag{4060} \tag{4061} \tag{4062} \tag{4063} \tag{4064} \tag{4065} \tag{4066} \tag{4067} \tag{4068} \tag{4069} \tag{4070} \tag{4071} \tag{4072} \tag{4073} \tag{4074} \tag{4075} \tag{4076} \tag{4077} \tag{4078} \tag{4079} \tag{4080} \tag{4081} \tag{4082} \tag{4083} \tag{4084} \tag{4085} \tag{4086} \tag{4087} \tag{4088} \tag{4089} \tag{4090} \tag{4091} \tag{4092} \tag{4093} \tag{4094} \tag{4095} \tag{4096} \tag{4097} \tag{4098} \tag{4099} \tag{4100} \right. f(x_2)

  • Decreasing Function Condition: A function $f$ is defined as decreasing if for all $x_1, x_2$ in its domain:     x_1 < x_2 \rightarrow f(x_1) \begin{matrix} \tag{4101} \tag{4102} \tag{4103} \tag{4104} \tag{4105} \tag{4106} \tag{4107} \tag{4108} \tag{4109} \tag{4110} \tag{4111} \tag{4112} \tag{4113} \tag{4114} \tag{4115} \tag{4116} \tag{4117} \tag{4118} \tag{4119} \tag{4120} \tag{4121} \tag{4122} \tag{4123} \tag{4124} \tag{4125} \tag{4126} \tag{4127} \tag{4128} \tag{4129} \tag{4130} \tag{4131} \tag{4132} \tag{4133} \tag{4134} \tag{4135} \tag{4136} \tag{4137} \tag{4138} \tag{4139} \tag{4140} \tag{4141} \tag{4142} \tag{4143} \tag{4144} \tag{4145} \tag{4146} \tag{4147} \tag{4148} \tag{4149} \tag{4150} \tag{4151} \tag{4152} \tag{4153} \tag{4154} \tag{4155} \tag{4156} \tag{4157} \tag{4158} \tag{4159} \tag{4160} \tag{4161} \tag{4162} \tag{4163} \tag{4164} \tag{4165} \tag{4166} \tag{4167} \tag{4168} \tag{4169} \tag{4170} \tag{4171} \tag{4172} \tag{4173} \tag{4174} \tag{4175} \tag{4176} \tag{4177} \tag{4178} \tag{4179} \tag{4180} \tag{4181} \tag{4182} \tag{4183} \tag{4184} \tag{4185} \tag{4186} \tag{4187} \tag{4188} \tag{4189} \tag{4190} \tag{4191} \tag{4192} \tag{4193} \tag{4194} \tag{4195} \tag{4196} \tag{4197} \tag{4198} \tag{4199} \tag{4200} \right. f(x_2)

• Monotonicity Proof Example:

  • Given function: $f(x) = 2x + 3$

  • Let $x_1 < x_2$

  • Evaluate function values: $f(x_1) = 2x_1 + 3$, $f(x_2) = 2x_2 + 3$

  • Inequality manipulation:     $x_1 < x_2 \rightarrow 2x_1 < 2x_2 \rightarrow 2x_1 + 3 < 2x_2 + 3$

  • Consequently: $f(x_1) < f(x_2)$

  • Conclusion: The function $f(x) = 2x + 3$ is strictly increasing.

Symmetry: Even and Odd Functions

• Even Functions (Symmetry about the Vertical Axis):

  • A function $f$ is symmetrical about the vertical axis (y-axis) when for all values of $x$ in the domain:     $f(-x) = f(x)$

  • Functions satisfying this condition are termed even functions.

  • Algebraic Example:     $f(x) = x^4 - 3x^2 + 4$

  • Step-by-Step Verification:     $f(-x) = (-x)^4 - 3(-x)^2 + 4 = x^4 - 3x^2 + 4 = f(x)$

  • Graphic illustration of an even function symmetrical about the vertical axis:     

    

• Odd Functions (Symmetry about the Origin):

  • A function $f$ is symmetrical about the origin if for all values of $x$ in the domain:     $f(-x) = -f(x)$

  • Functions satisfying this condition are termed odd functions.

  • Algebraic Example:     $f(x) = x^5 - 3x^3$

  • Step-by-Step Verification:     $f(-x) = (-x)^5 - 3(-x)^3 = -x^5 + 3x^3 = -(x^5 - 3x^3) = -f(x)$

  • Graphic illustration of an odd function symmetrical about the origin:     

    

Domain Determination Across Function Types

• Definitions of Domain and Range:

  • The domain is the set of elements that possesses an image.

  • In mathematical terms, the set of image values is termed the range.

  • Different function types exhibit specific domain and range characteristics.

• Domain of a Polynomial Function:

  • The domain is $R$ (all real numbers).

  • Example: For $f(x) = x^2 - 5x + 6$, the domain is $D = R$.

• Domain of a Rational Function:

  • The domain is $R$, minus any values that annul the denominator (since division by zero is undefined).

  • Example: For a denominator $x^2 - 5x + 6 = 0$:     $(x - 2)(x - 3) = 0 \rightarrow x = 2, x = 3$

  • Domain statement: $D = R - \{2, 3\}$.

• Domain of an Irrational Function of Odd Index:

  • The domain is $R$ (all real numbers).

  • Example 1 (Pure radical): $f(x) = \sqrt[3]{x^2 - 5x + 6} \rightarrow D = R$

  • Example 2 (Odd radical with rational interior): $f(x) = \sqrt[3]{\frac{x}{x^2 - 5x + 6}} \rightarrow D = R - \{2, 3\}$

• Domain of an Irrational Function of Even Index:

  • The domain comprises all values that make the radicand greater than or equal to zero (\text{radicand} \begin{matrix} \tag{4201} \tag{4202} \tag{4203} \tag{4204} \tag{4205} \tag{4206} \tag{4207} \tag{4208} \tag{4209} \tag{4210} \tag{4211} \tag{4212} \tag{4213} \tag{4214} \tag{4215} \tag{4216} \tag{4217} \tag{4218} \tag{4219} \tag{4220} \tag{4221} \tag{4222} \tag{4223} \tag{4224} \tag{4225} \tag{4226} \tag{4227} \tag{4228} \tag{4229} \tag{4230} \tag{4231} \tag{4232} \tag{4233} \tag{4234} \tag{4235} \tag{4236} \tag{4237} \tag{4238} \tag{4239} \tag{4240} \tag{4241} \tag{4242} \tag{4243} \tag{4244} \tag{4245} \tag{4246} \tag{4247} \tag{4248} \tag{4249} \tag{4250} \tag{4251} \tag{4252} \tag{4253} \tag{4254} \tag{4255} \tag{4256} \tag{4257} \tag{4258} \tag{4259} \tag{4260} \tag{4261} \tag{4262} \tag{4263} \tag{4264} \tag{4265} \tag{4266} \tag{4267} \tag{4268} \tag{4269} \tag{4270} \tag{4271} \tag{4272} \tag{4273} \tag{4274} \tag{4275} \tag{4276} \tag{4277} \tag{4278} \tag{4279} \tag{4280} \tag{4281} \tag{4282} \tag{4283} \tag{4284} \tag{4285} \tag{4286} \tag{4287} \tag{4288} \tag{4289} \tag{4290} \tag{4291} \tag{4292} \tag{4293} \tag{4294} \tag{4295} \tag{4296} \tag{4297} \tag{4298} \tag{4299} \tag{4300} \end{matrix} 0).

  • Example 1: $f(x) = \sqrt{x^2 - 5x + 6}$

  • Solve inequality: x^2 - 5x + 6 \begin{matrix} \tag{4301} \tag{4302} \tag{4303} \tag{4304} \tag{4305} \tag{4306} \tag{4307} \tag{4308} \tag{4309} \tag{4310} \tag{4311} \tag{4312} \tag{4313} \tag{4314} \tag{4315} \tag{4316} \tag{4317} \tag{4318} \tag{4319} \tag{4320} \tag{4321} \tag{4322} \tag{4323} \tag{4324} \tag{4325} \tag{4326} \tag{4327} \tag{4328} \tag{4329} \tag{4330} \tag{4331} \tag{4332} \tag{4333} \tag{4334} \tag{4335} \tag{4336} \tag{4337} \tag{4338} \tag{4339} \tag{4340} \tag{4341} \tag{4342} \tag{4343} \tag{4344} \tag{4345} \tag{4346} \tag{4347} \tag{4348} \tag{4349} \tag{4350} \tag{4351} \tag{4352} \tag{4353} \tag{4354} \tag{4355} \tag{4356} \tag{4357} \tag{4358} \tag{4359} \tag{4360} \tag{4361} \tag{4362} \tag{4363} \tag{4364} \tag{4365} \tag{4366} \tag{4367} \tag{4368} \tag{4369} \tag{4370} \tag{4371} \tag{4372} \tag{4373} \tag{4374} \tag{4375} \tag{4376} \tag{4377} \tag{4378} \tag{4379} \tag{4380} \tag{4381} \tag{4382} \tag{4383} \tag{4384} \tag{4385} \tag{4386} \tag{4387} \tag{4388} \tag{4389} \tag{4390} \tag{4391} \tag{4392} \tag{4393} \tag{4394} \tag{4395} \tag{4396} \tag{4397} \tag{4398} \tag{4399} \tag{4400} \end{matrix} 0

  • Domain solution: $D = (-\infty, 2] \cup [3, \infty)$

  • Example 2 (Combination of even radical numerator and linear denominator): $f(x) = \frac{\sqrt{x^2 - 5x + 6}}{x + 4}$

  • Numerator condition: x^2 - 5x + 6 \begin{matrix} \tag{4401} \tag{4402} \tag{4403} \tag{4404} \tag{4405} \tag{4406} \tag{4407} \tag{4408} \tag{4409} \tag{4410} \tag{4411} \tag{4412} \tag{4413} \tag{4414} \tag{4415} \tag{4416} \tag{4417} \tag{4418} \tag{4419} \tag{4420} \tag{4421} \tag{4422} \tag{4423} \tag{4424} \tag{4425} \tag{4426} \tag{4427} \tag{4428} \tag{4429} \tag{4430} \tag{4431} \tag{4432} \tag{4433} \tag{4434} \tag{4435} \tag{4436} \tag{4437} \tag{4438} \tag{4439} \tag{4440} \tag{4441} \tag{4442} \tag{4443} \tag{4444} \tag{4445} \tag{4446} \tag{4447} \tag{4448} \tag{4449} \tag{4450} \tag{4451} \tag{4452} \tag{4453} \tag{4454} \tag{4455} \tag{4456} \tag{4457} \tag{4458} \tag{4459} \tag{4460} \tag{4461} \tag{4462} \tag{4463} \tag{4464} \tag{4465} \tag{4466} \tag{4467} \tag{4468} \tag{4469} \tag{4470} \tag{4471} \tag{4472} \tag{4473} \tag{4474} \tag{4475} \tag{4476} \tag{4477} \tag{4478} \tag{4479} \tag{4480} \tag{4481} \tag{4482} \tag{4483} \tag{4484} \tag{4485} \tag{4486} \tag{4487} \tag{4488} \tag{4489} \tag{4490} \tag{4491} \tag{4492} \tag{4493} \tag{4494} \tag{4495} \tag{4496} \tag{4497} \tag{4498} \tag{4499} \tag{4500} \end{matrix} 0 \rightarrow (-\infty, 2] \cup [3, \infty)

  • Denominator condition: $x + 4 \neq 0 \rightarrow x \neq -4$

  • Combined domain solution: $D = (-\infty, -4) \cup (-4, 2] \cup [3, \infty)$

• Domain of a Logarithmic Function:

  • The domain comprises all values that make the inside argument strictly greater than zero ($\text{inside} > 0$).

  • Example: For $x^2 - 5x + 6 > 0$

  • Domain solution: $D = (-\infty, 2) \cup (3, \infty)$

• Domain of Exponential and Trigonometric Functions:

  • Exponential Function: Domain is $R$

  • Sine Function: Domain is $R$

  • Cosine Function: Domain is $R$

Composition and Inverse Functions

• Composition of Functions:

  • Definition: Given two functions $f(x)$ and $g(x)$, where the domain of the second function lies within the range of the first, the composite function associates each element of the domain of $f(x)$ with the value $g[f(x)]$.

  • Operational Example:     $f(x) = 3x + 2, \quad g(x) = \frac{x + 3}{2x + 1}$

  • Derivation of $g \circ f = g[f(x)]$:     $g[f(x)] = g(3x + 2) = \frac{(3x + 2) + 3}{2(3x + 2) + 1} = \frac{3x + 5}{6x + 5}$

  • Derivation of $f \circ g = f[g(x)]$:     $f[g(x)] = f\left(\frac{x + 3}{2x + 1}\right) = 3\left(\frac{x + 3}{2x + 1}\right) + 2 = \frac{3(x + 3) + 2(2x + 1)}{2x + 1} = \frac{7x + 11}{2x + 1}$

• Inverse Function Determination:

  • Procedure:

  • Step 1: Replace the function notation $f(x)$ with $y$.

  • Step 2: Rearrange the equation to express $x$ explicitly in terms of $y$.

  • Step 3: Replace $x$ with $f^{-1}(x)$ and replace $y$ with x$.\n * Worked Example:\n * Given: f(x) = \frac{2x + 1}{3x}\n * Step 1: Set y = \frac{2x + 1}{3x}\n * Step 2: Solve for x:\n      y \times 3x = 2x + 1\n      3xy = 2x + 1\n      3xy - 2x = 1\n      x(3y - 2) = 1\n      x = \frac{1}{3y - 2}\n * Step 3: Substitute notation:\n      f^{-1}(x) = \frac{1}{3x - 2}\n\n# Quadratic Functions: Extrema and Vertices\n\n* Standard Quadratic Form:\n * A quadratic function is expressed as: f(x) = ax^2 + bx + c$, where$a eq 0\n * The vertex corresponds to the absolute maximum or minimum point of the parabola.\n\n* Orientation and Extrema Rules:\n * If a > 0: The parabola opens upwards, making the vertex a minimum point.\n * If a < 0: The parabola opens downwards, making the vertex a maximum point.\n\n* Vertex Coordinates Formula:\n * Horizontal coordinate: x_0 = -\frac{b}{2a}\n * Vertical coordinate: y_0 = f(x_0)\n\n* Worked Example:\n * Problem: Determine the minimum value of y = 5x^2 + 20x + 9.\n * Step 1: Calculate x_0:\n    x_0 = -\frac{20}{2 \cdot 5} = -\frac{20}{10} = -2\n * Step 2: Calculate y_0 = f(x_0):\n    f(-2) = 5(-2)^2 + 20(-2) + 9 = 5(4) - 40 + 9 = 20 - 40 + 9 = -11\n * Result: The function reaches a minimum value of -11$when$x = -2\n * Graphic illustration of vertex minimum:\n    \n\n# Questions and Practice Problems\n\n* Question 1:\n * Prompt: If f(x) = \log_2(x^2 + 12)$, what is the reciprocal of$f(2)?\n * Options:\n * a) \frac{1}{4}\n * b) 4\n * c) 2\n * d) \frac{1}{2}\n * e) 6\n * Solution Detail: Evaluate f(2) = \log_2(2^2 + 12) = \log_2(16) = 4$. The reciprocal of$4$is$\frac{1}{4}. Correct choice is a).\n\n* Question 2:\n * Prompt: What are the coordinates of the turning point of the graph of the quadratic equation y = 2x^2 + 5x - 2?\n * Options:\n * a) \left(-\frac{5}{4}, -\frac{41}{8}\right)\n * b) \left(-\frac{5}{2}, -\frac{41}{16}\right)\n * c) \left(\frac{5}{4}, -\frac{41}{16}\right)\n * d) \left(-\frac{5}{2}, -\frac{41}{8}\right)\n * e) \left(-\frac{5}{4}, -\frac{41}{16}\right)\n * Solution Detail: x_0 = -\frac{5}{2(2)} = -\frac{5}{4}$. Evaluating$y_0 = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) - 2 = \frac{25}{8} - \frac{25}{4} - 2 = -\frac{41}{8}. Correct choice is a).\n\n* Question 3:\n * Prompt: Consider this function, defined for every x > 1$(where '$\ln$' indicates the natural logarithm):$f(x) = 2\ln(x) - 2\ln(x - 1)$. Which of these is an expression for its inverse function$f^{-1}(y)?\n * Options:\n * a) \frac{e^{y/2}}{e^{y/2}-1}, y > 1\n * b) \frac{1}{e^{y/2}-1}, y > 0\n * c) \frac{1}{e^{y/2}-1}, y > 1\n * d) \frac{e^{y/2}}{e^{y/2}-1}, y > 0\n * e) \frac{1}{e^{y/2}-1}, y eq 0\n * Solution Detail: Rewrite as y = 2\ln\left(\frac{x}{x-1}\right) \rightarrow \frac{y}{2} = \ln\left(\frac{x}{x-1}\right) \rightarrow e^{y/2} = \frac{x}{x-1}$. Solving for$x$yields$x = \frac{e^{y/2}}{e^{y/2}-1}$. Since$x > 1$,$\frac{x}{x-1} > 1$, meaning$y > 0. Correct choice is d).\n\n* Question 4:\n * Prompt: The maximum value assumed by the function y = -x^2 + 5x - 4 is:\n * Options:\n * a) +\infty\n * b) \frac{9}{4}\n * c) 4\n * d) 1\n * e) \frac{5}{2}\n * Solution Detail: Vertex position x_0 = -\frac{5}{2(-1)} = \frac{5}{2}$. Maximum value$y_{max} = -\left(\frac{5}{2}\right)^2 + 5\left(\frac{5}{2}\right) - 4 = -\frac{25}{4} + \frac{25}{2} - 4 = \frac{9}{4}. Correct choice is b).\n\n* Question 5:\n * Prompt: Which is the max value of the function f(x) = 4x - x^2?\n * Options:\n * a) 0\n * b) 2\n * c) 4\n * d) 6\n * e) 8\n * Solution Detail: x_0 = -\frac{4}{2(-1)} = 2$. Maximum value$f(2) = 4(2) - (2)^2 = 4. Correct choice is c).\n\n* Question 6:\n * Prompt: Which one of the following is the composite function f \circ g(x)$where$f(x) = 2x - 1$and$g(x) = 6 - x?\n * Options:\n * a) f \circ g(x) = 13 - 2x\n * b) f \circ g(x) = 11 - x\n * c) f \circ g(x) = 11 - 2x\n * d) f \circ g(x) = 7 - 2x\n * e) f \circ g(x) = 5 - 2x\n * Solution Detail: f(g(x)) = 2(6 - x) - 1 = 12 - 2x - 1 = 11 - 2x. Correct choice is c).\n\n* Question 7:\n * Prompt: Which function below is an odd function?\n * Options:\n * a) f(x) = x^3 + x^2\n * b) f(x) = 4x^3 + 2x\n * c) f(x) = 2x^2 - 8x\n * d) f(x) = 3x^4 + 5x^2\n * e) None of the above\n * Solution Detail: Check choice b: f(-x) = 4(-x)^3 + 2(-x) = -4x^3 - 2x = -(4x^3 + 2x) = -f(x). Correct choice is b).\n\n* Question 8:\n * Prompt: Let the function f$be defined by$f(x) = \frac{2x + 1}{x - 4}, x eq 4$. Find the value of$f^{-1}(x).\n * Options:\n * a) 5\n * b) 13\n * c) 0\n * d) -5\n * e) -13\n\n* Question 9:\n * Prompt: Find the domain of the function: f(x) = \frac{\sqrt{x^2 - 4}}{x - 1}\n * Options:\n * a) (-\infty, -2] \cup [2, +\infty)\n * b) (-\infty, -2] \cup [2, +\infty) - {1}\n * c) [-2, 2] - {1}\n * d) (-2, 2) - {1}\n * e) (-\infty, -2) \cup (2, +\infty)\n * Solution Detail: Radicand condition x^2 - 4 \begin{matrix} \tag{4501} \tag{4502} \tag{4503} \tag{4504} \tag{4505} \tag{4506} \tag{4507} \tag{4508} \tag{4509} \tag{4510} \tag{4511} \tag{4512} \tag{4513} \tag{4514} \tag{4515} \tag{4516} \tag{4517} \tag{4518} \tag{4519} \tag{4520} \tag{4521} \tag{4522} \tag{4523} \tag{4524} \tag{4525} \tag{4526} \tag{4527} \tag{4528} \tag{4529} \tag{4530} \tag{4531} \tag{4532} \tag{4533} \tag{4534} \tag{4535} \tag{4536} \tag{4537} \tag{4538} \tag{4539} \tag{4540} \tag{4541} \tag{4542} \tag{4543} \tag{4544} \tag{4545} \tag{4546} \tag{4547} \tag{4548} \tag{4549} \tag{4550} \tag{4551} \tag{4552} \tag{4553} \tag{4554} \tag{4555} \tag{4556} \tag{4557} \tag{4558} \tag{4559} \tag{4560} \tag{4561} \tag{4562} \tag{4563} \tag{4564} \tag{4565} \tag{4566} \tag{4567} \tag{4568} \tag{4569} ag{4570} \tag{4571} \tag{4572} \tag{4573} \tag{4574} \tag{4575} \tag{4576} \tag{4577} \tag{4578} \tag{4579} \tag{4580} \tag{4581} \tag{4582} \tag{4583} \tag{4584} \tag{4585} \tag{4586} \tag{4587} \tag{4588} \tag{4589} \tag{4590} \tag{4591} \tag{4592} \tag{4593} \tag{4594} \tag{4595} \tag{4596} \tag{4597} \tag{4598} \tag{4599} \tag{4600} \end{matrix} 0 \rightarrow (-\infty, -2] \cup [2, +\infty)$. Denominator condition$x - 1 eq 0 \rightarrow x eq 1$. Since$1$is outside$(-\infty, -2] \cup [2, +\infty)$, removing$1$$ does not alter the set. Correct choice is a).